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UNIVERSITY  OF  CALIFORNIA. 


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MANUAL  li  Library. 

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SPHERICAL  AND  PRACTICAL 

ASTRONOMY: 


EMBRACING 

THE  GENERAL  PROBLEMS  OF  SPHERICAL  ASTRONOMY,  THE  SPECIAL 
APPLICATIONS  TO  NAUTICAL  ASTRONOMY,  AND  THE  THEORY 
AND  USE  OF  FIXED  AND  PORTABLE  ASTRO- 
NOMICAL INSTRUMENTS. 


WITH  AN  APPENDIX   ON   THE    METHOD   OF   LEAST    SQUARES, 


BY 

WILLIAM  CHAUVEKET, 

PROFESSOR    OF    MATHEMATICS    AND    ASTRONOMY    IN    WASHINGTON    UNIVERSITY,    SAINT    LOCIS. 


VOL.  I. 
SPHEKICAL  ASTRONOMY. 

FIFTH  EDITION,   REVISED   AND  CORRECTED. 

PHILADELPHIA: 

J.    B.    L  I  P  P  I  lyC  O  T  T    &    CO. 

LONDON:  TBfUBNEK  &  CO. 

1«74 


M.   \ 


Entered,  according  to  Act  of  Congress,  in  the  year  1863,  by 

J.  V.  LIPPINCOTT  &  CO. 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Eastern 
District  of  Pennsylvania. 


PREFACE. 


THE  methods  of  investigation  adopted  in  this  work  are 
in  accordance  with  what  may  be  called  the  modern  school 
of  practical  astronomy,  or  more  distinctively  the  Ger- 
man school,  at  the  head  of  which  stands  the  unrivalled 
BESSEL.  In  this  school,  the  investigations  both  of  the 
general  problems  of  Spherical  Astronomy  and  of  the  Theory 
of  Astronomical  Instruments  are  distinguished  by  the  gene- 
rality of  their  form  and  their  mathematical  rigor.  When 
approximative  methods  are  employed  for  convenience  in 
practice,  their  degree  of  accuracy  is  carefully  determined  by 
means  of  exact  formulae  previously  investigated ;  the  latter 
being  developed  in  converging  series,  and  only  such  terms 
of  these  series  being  neglected  as  can  be  shown  to  be  insen- 
sible in  the  cases  to  which  the  formulae  are  to  be  applied. 
A.nd  it  is  an  essential  condition  of  all  the  methods  of  com- 
putation from  data  furnished  by  observation,  that  the  errors 
of  the  computation  shall  always  be  practically  insensible  in 
relation  to  the  errors  of  observation  :  so  that  our  results 
shall  be  purely  the  legitimate  deductions  from  the  observa- 
tions, and  free  from  all  avoidable  error. 

It  is  another  characteristic  feature  of  modern  spherical 
astronomy,  that  the  final  formulae  furnished  to  the  practical 
computer  are  so  presented  as  seldom  to  require  accompany- 
ing verbal  precepts  to  distinguish  the  species  of  the  unknown 
angles  and  arcs ;  and  this  results,  in  a  great  measure,  from 
the  consideration  of  the  general  spherical  triangle,  or  that  in 
which  the  six  parts  of  the  triangle  are  not  subjected  to  the 

74V' 


PKEFACE. 


condition  that  they  shall  each  be  less  than  ISO0,  but  may 
have  any  values  less  than  36 0°,  all  ambiguity  as  to  their 
species  being  removed  by  determining  them,  when  necessary, 
by  two  of  their  trigonometric  functions,  usually  the  sine  and 
the  cosine.  This  feature  is  mainly  due  to  GAUSS,  and  was 
prominently  exhibited  in  his  Theoria  Mot  as .  Corporum  Coe- 
le-stium,  published  in  1809.  The  English  and  American 
astronomers  have  been  slow  to  adopt  this  manifest  improve- 
ment ;  in  evidence  of  which  I  may  remark  that  the  general 
spherical  triangle  was  not  treated  of  in  any  work  in  the 
English  language,  so  far  as  I  know,  prior  to  the  publication 
of  my  Treatise  on  Plane  and  Spherical  Trigonometry,  in  the 
year  1850.  In  the  present  work,  I  assume  the  reader  to  be 
acquainted  with  this  form  of  spherical  trigonometry,  and  to 
accept  its  fundamental  equations  in  their  utmost  generality. 

A  third  and  eminently  characteristic  feature  of  modern 
astronomy,  is  the  use  which  it  makes,  in  all  its  departments, 
of  the  method  of  least  squares,  namely,  that  method  of 
combining  observations  which  shall  give  the  most  probable 
results,  or  which  shall  be  exposed  to  the  least  probable  errors. 
This  method  is  also  due  to  GAUSS,  who  (though  anticipated 
in  the  publication  of  one  of  its  practical  rules  by  LEGENDRE) 
was  the  first  to  give  a  philosophical  exposition  of  its  princi- 
ples. The  direct  effect  of  this  improvement  is  not  only  that 
the  most  probable  result  in  each  case  is  obtained,  but  also 
that  the  relative  degree  of  accuracy  of  that  result  is  deter- 
mined, and  thus  the  degree  of  confidence  with  which  it  may 
be  received  and  the  weight  which  it  may  be  allowed  to  have 
in  subsequent  discussions.  Judiciously  employed,  it  serves 
to  indicate  when  a  particular  process  has  reached  the  limit 
of  accuracy  which  it  can  afford,  thereby  saves  fruitless 
labor,  directs  inquiry  into  new  channels,  and  contributes 
greatly  to  accelerate  the  progress  of  the  science. 

Whilst  the  science  has  been  rapidly  advancing  in  Europe, 
we  have  in  this  country  not  been  idle.  Two  of  the  most 
important  improvements  in  practical  astronomy  have  had 


PREFACE. 


their  origin  in  the  United  States, — the  method  of  finding 
differences  of  geographical  longitude  by  the  electric  telegraph, 
and  that  of  finding  the  geographical  latitude  by  the  zenith 
telescope.  These  are  the  direct  offspring  of  our  admirably 
conducted  Coast  Survey,  which,  with  the  aid  of  these 
methods,  both  of  the  greatest  simplicity,  has  fixed  the  lati- 
tudes and  relative  longitudes  of  a  series  of  points  on  our 
coast  with  a  degree  of  accuracy  wholly  unapproached  in  any 
previous  work  of  this  kind.  This  extreme  accuracy  will  be 
apparent  to  the  reader  who  will  refer  to  the  examples  here 
given,  which  have  been  selected  (almost  at  random)  from 
the  records  of  the  Survey. 

It  is  perhaps  necessary  to  say  a  few  words  here  respect- 
ing those  portions  of  this  treatise  in  which  I  have  ventured 
to  substitute  my  own  methods  for  those  heretofore  employed. 
My  method  of  reducing  lunar  distances,  which  was  first 
published  in  the  American  Ephemeris  for  1855,  is  here  re- 
produced, together  with  the  necessary  tables  for  its  applica- 
tion. But  I  have  first,  for  the  sake  of  completeness,  given 
the  usual  rigorous  solution,  although  this  is  confessedly  too 
laborious  for  ordinary  use,  and  especially  for  use  at  sea.  The 
approximative  methods  heretofore  proposed  may  be  divided 
into  two  classes :  first,  those  based  upon  sufficiently  precise 
formulae,  but  such  that  the  tables  required  in  their  applica- 
tion are  adapted  only  to  a  mean  state  of  the  atmosphere ; 
and  second,  those  based  upon  incomplete  formulas.  As  to 
the  first  class,  the  trouble  of  correcting  the  tabular  numbers 
for  the  barometer  and  thermometer  would  render  the 
methods  as  laborious  as  the  rigorous  method,  and  it  is 
therefore  the  usual  practice,  at  sea,  to  disregard  these  correc- 
tions altogether,  thus  introducing  a  greater  error  than  would 
follow  from  the  use  of  the  more  incomplete  formulae  of  the 
second  class,  if  in  the  latter  these  corrections  were  taken 
into  account.  But,  as  to  the  methods  of  the  second  class  (of 
which  there  are  several  in  common  use),  it  will  be  found 
upon  examination  that  the  omitted  terms  of  the  formula} 


6  PREFACE. 

are  not  so  small  as  to  be  insensible  even  in  relation  to  the 
rather  large  errors  of  observation  which  are  unavoidable  IP 
the  use  of  the  sextant.  The  defects  of  both  classes  are 
supposed  to  be  avoided  in  my  new  method ;  for,  first,  I  have 
deduced  a  rigorous  formula  from  which  is  derived  an  ap- 
proximate one,  practically  perfect,  representing  the  true  cor- 
rection of  the  lunar  distance  within  one  second  of  arc  in 
every  case  that  can  occur  in  practice ;  and,  second,  I  have 
arranged  this  formula  so  that  it  not  only  requires  extremely 
simple  tables  in  its  application,  but  also  the  tabular  numbers 
require  no  correction  for  the  barometer  and  thermometer,  the 
corrections  for  the  state  of  these  instruments  being  intro- 
duced in  a  simple  manner  in  forming  the  arguments  of  the 
tables.  In  applying  this  method  with  logarithms  of  only 
four  decimal  places,  the  true  distance  is  usually  obtained 
within  less  than  two  seconds  of  arc,  a  degree  of  accuracy  far 
greater  than  is  necessary  in  relation  to  our  present  means 
of  observing  the  distance.  It  is,  in  fact,  quite  as  accurate 
in  practice  as  BESSEL'S  theoretically  exact  method  when  the 
latter  is  also  carried  out  with  four-place  logarithms.  I 
think,  therefore,  that  I  may  justly  prefer  my  own  method 
not  only  to  the  imperfect  approximative  methods  above 
referred  to,  but  also  to  BESSEL'S  method,  which  requires  an 
extended  Ephemeris  wholly  different  from  that  now  in  use, 
and  is  withal  more  laborious. 

The  Gaussian  method  of  reducing  circummeridian  alti- 
tudes of  the  sun  by  referring  them  to  the  instant  of  the 
sun's  maximum  altitude,  is  in  this  work  rigorously  investi- 
gated, and  a  small  term,  overlooked  or  disregarded  by  GAUSS, 
has  been  added  to  the  formula. 

A  new  and  brief  approximative  method  of  finding  the 
latitude  by  two  altitudes  near  the  meridian  when  the 
time  is  not  known,  is  given  in  Vol.  I.  Arts.  195  and  204,  and 
another  by  three  altitudes  near  the  meridian,  in  Art.  205, 
which  will  probably  be  found  useful  as  nautical  methods. 

The  subject  of  Eclipses  will  be  found  treated  with  more 


PREFACE.  7 

than  usual  completeness.  The  fundamental  formulae  adopted 
are  those  of  BESSEL'S  theory,  but  the  solutions  of  the  various 
problems  relating  to  the  prediction  of  solar  eclipses  for  the 
earth  generally  are  mostly  new.  The  rigorous  solutions  of 
these  problems  given  by  BESSEL  in  his  Analyse  der  Finster- 
nesse  are  not  required  for  the  usual  purposes  of  prediction, 
however  interesting  they  may  be  as  specimens  of  refined 
and  elegant  analysis.  On  the  other  hand,  the  approximate 
solutions  commonly  given  appear  to  be  unnecessarily  rude. 
Those  that  I  have  substituted  will  be  found  to  be  very  little 
if  at  all  more  laborious  than  the  latter,  while  they  are  almost 
as  precise  as  the  former,  and  by  a  very  little  additional  labor 
(that  is,  by  repeating  only  some  parts  of  the  computation 
for  a  second  or  third  approximation)  may  be  rendered  quite 
exact. 

So  far  as  I  can  find,  no  one  has  heretofore  treated  distinct- 
ively of  the  occultations  of  planets  by  the  moon,  and  these 
phenomena  have  been  dismissed  as  simple  cases  of  the 
general  theory  of  eclipses,  in  which  both  the  occulting  and 
the  occulted  body  are  spherical.  But  in  almost  every  oc- 
cultation  of  one  of  the  principal  planets,  the  planet  will  be 
either  a  spheroidal  body  fully  or  partially  illuminated  by 
the  sun,  or  a  spherical  body  partially  illuminated :  so  that, 
in  the  general  case,  we  have  to  consider  the  disc  of  the  oc- 
culted body  as  bounded  by  an  ellipse  or  by  two  different 
semi-ellipses.  I  have  discussed  this  general  case  at  length, 
and  have  adapted  the  theory  to  each  planet  specially.  The 
additional  computations  required  to  take  into  account  the 
true  figure  of  the  planet's  disc  are  sufficiently  brief  and 
simple.  The  case  of  the  occultation  of  a  cusp  of  Yenus  or 
Mercury  is  included  in  the  discussion,  and  also  the  occulta- 
tion of  Saturn's  rings. 

The  well  known  formula  for  predicting  the  transits  of  the 
inferior  planets  over  the  sun's  disc,  first  given  by  LAGRANGE, 
is  here  rendered  more  accurate  by  introducing  a  considera- 


8  PREFACE. 

tion  of  the  compression  of  the  earth ;  and  a  new  and  simple 
demonstration  of  the  formula  is  given. 

In  the  practical  portions  of  the  work,  and  especially  in 
the  second  volume,  I  have  endeavored  to  give  every  import/ 
ant  precept  for  the  guidance  of  observers,  deduced  from  the 
labors  of  others  or  suggested  by  my  own  experience.  All 
the  principal  methods  are  illustrated  by  examples  from 
actual  observation. 

I  have  taken  especial  pains  throughout  the  work  to  ex- 
hibit the  mode  of  discussing  the  probable  errors  of  the  results 
obtained  by  observations,  and  have  given  numerous  examples 
of  the  application  of  the  method  of  least  squares.  This 
method  is  applicable  in  almost  all  the  physical  sciences 
where  numerical  results  are  to  be  deduced,  and,  therefore, 
does  not  necessarily  form  a  part  of  a  work  on  astronomy ; 
but,  as  I  could  not  refer  my  reader  to  any  work  in  the 
English  language  for  a  sufficient  account  of  the  method,  I 
have  prepared  a  concise  treatise  upon  it,  which  forms  the 
Appendix.  In  this,  I  have  confined  myself  chiefly  to  the 
parts  of  the  theory  required  in  practical  astronomy,  and  have 
endeavored  to  present  its  principles  in  a  simple  yet  rigorous 
manner  (so  far  as  the  subject  allows),  taking  as  a  basis 
known  theorems  of  the  calculus  of  probabilities,  and  follow- 
ing principally  the  processes  first  proposed  by  GAUSS. 

In  this  Appendix  I  have  treated  of  PEIRCE'S  Criterion  for 
the  rejection  of  doubtful  observations,  which  is  already  well 
known  to  American  astronomers,  and  is  now  constantly 
applied  in  the  discussion  of  observations  upon  our  Coast 
Survey.  Objections  have  been  made  to  the  criterion,  but 
none  that  would  not  apply  equally  well  to  the  method  of 
least  squares  itself.  To  those  who  have  not  been  able  to 
follow  PEIRCE'S  investigation,  the  simple  approximate  cri- 
terion which  I  have  suggested  at  the  end  of  the  Appendix 
may  prove  acceptable.  It  is  derived  directly  from  the  fun- 
damental formula  of  the  method  of  least  squares,  and  leads 


PREFACE. 

to  the  rejection  of  nearly  the  same  observations  as  that  of 
PEIRCE. 

The  plates  at  the  end  of  the  work  exhibit  in  minute 
detail  the  instruments  now  chiefly  employed  by  astronomers. 
To  have  given  more,  with  the  necessary  explanations,  would 
have  led  me  too  far  into  the  mere  history  of  the  subject,  and 
would  have  occupied  space  which  I  thought  it  preferable  to 
fill  with  discussions  relating  to  the  leading  instruments  now 
in  use.  The  scale  of  these  plates  is  purposely  made  quite 
small;  but  the  great  precision  with  w^hich  they  are  executed 
will  enable  the  reader  to  measure  from  them  the  dimensions 
of  all  the  important  parts  of  each  of  the  principal  instru- 
ments. I  am  greatly  indebted  for  the  perfection  of  these 
drawings  to  the  engravers,  the  Messrs.  ILLMAN  BROTHERS,  of 
Philadelphia. 

Such  auxiliary  tables  as  seemed  to  be  necessary  to  the 
reader  in  using  these  volumes  have  been  given  at  the  end 
of  Vol.  II.  Some  of  these  are  new.  Most  of  those  which 
have  been  derived  from  other  sources  have  been  either  re- 
computed or  tested  by  differences  and  corrected.  To  insure 
their  accuracy,  they  have  also  been  tested  by  differences 
after  being  in  type. 

For  the  very  complete  index  to  the  whole  work,  I  am 
indebted  to  my  friend,  Prof.  J.  D.  CREHORE,  of  Washington 
University. 

In  conclusion,  I  desire  to  express  my  obligations  to  those 
citizens  of  Saint  Louis  who,  without  solicitation,  have  gene- 
rously assumed  a  share  of  the  risk  of  publication.  Their 
liberal  spirit  has  been  met  by  a  corresponding  liberality  on 
the  part  of  my  publishers,  who  have  spared  no  expense  in 
the  typographical  execution.  I  shall  be  content  if  their 
expectations  are  not  wholly  disappointed,  and  the  work 
contributes  in  any  degree  to  the  advancement  of  the  noblest 
of  the  physical  sciences. 

WASHINGTON  UNIVERSITY, 

SAINT  Louis,  January  1,  1863. 


CONTENTS  OF  VOL.  I. 


SPHEEICAL  ASTRONOMY. 


CHAPTER  I. 

PAGK 

THE  CELESTIAL  SPHERE — SPHERICAL  AND  RECTANGULAR  CO-ORDINATES 17 

Spherical  co-ordinates 18 

Transformation  of  spherical  co-ordinates 27 

Rectangular  co-ordinates. .. 43 

Transformation  of  rectangular  co-ordinates 48 

Differential  variations  of  co-ordinates...  50 


CHAPTER  II. 

TIME — USE  OF  THE  EPHEMERIS — INTERPOLATION — STAR  CATALOGUES 52 

Solar  time 53 

Sidereal  time 59 

Hour  angles 64 

Ephemeris 68 

Interpolation  by  differences  of  any  order 79 

Star  catalogues 91 

CHAPTER  III. 

FIGURE  AND  DIMENSIONS  OF  THE  EARTH 95 

Reduction  of  latitude » 97 

Radius  of  the  terrestrial  spheroid  for  given  latitudes 99 

Normal,  &c 101 

CHAPTER  IV. 

REDUCTION  OF  OBSERVATIONS  TO  THE  CENTRE  OF  THE  EARTH 103 

Parallax 104 

Refraction. — General  laws  of  refraction 127 

Tables  of  refraction 130 

Differential  equation  of  the  atmospheric  refraction 136 

Integration  of  the  differential  equation  with  BOUGUER'S  hypothesis 136 

Integration  with  BESSEL'S  hypothesis  according  to  the  methods  of  KRAMP 

and  LAPLACE 143 

11 


CONTENTS. 

PAOE 

Construction  of  BESSEL'S  Table 105 

Refraction  in  right  ascension  and  declination 171 

Dip  of  the  horizon 172 

Semidiameters  of  celestial  bodies 180 

Augmentation  of  the  moon's  semidiameter 183 

Contraction  of  the  sun's  and  the  moon's  semidiameters  by  refraction 184 

Reduction  of  observed  zenith  distances  to  the  centre  of  the  earth 189 


CHAPTER  V. 

FINDING  THE  TIME  BY  ASTRONOMICAL  OBSERVATIONS 193 

1st  Method.— By  transits.... 196 

2d   Method.— By  equal  altitudes 196 

3d  Method. — By  a  single  altitude  or  zenith  distance 206 

Correction  for  second  differences  of  zenith  distance 213 

4th  Method. — By  the  disappearance  of  a  star  behind  a  terrestrial  object....   217 

Time  of  rising  and  setting  of  the  stars 218 

FINDING  THE  TIME  AT  SEA 219 

1st  Method.— By  a  single  altitude ..    219 

2d  Method. — By  equal  altitudes 220 

CHAPTER  VI. 

FINDING  THE  LATITUDE  BY  ASTRONOMICAL  OBSERVATIONS 223 

1st  Method. — By  meridian  altitudes  or  zenith  distances 223 

Combination  of  pairs  of   stars  whose    meridian  zenith    distances   are 

nearly  equal  (see  Vol.  II.,  Zenith  Telescope)  226 

Meridian  altitudes  of  a  circumpolar  star 226 

Meridian  zenith  distances  of  the  sun  near  the  solstices 228 

2d  Method. — By  a  single  altitude  at  a  given  time 229 

3d  Method. — By  reduction  to  the  meridian  when  the  time  is  given 233 

Circummeridian  altitudes 235 

GAUSS'S  method  of  reducing  circummeridian  altitudes  of  the  sun 244 

Limits  of  the  reduction  to  the  meridian 251 

4th  Method.— By  the  Pole  Star 253 

5th  Method. — By  two  altitudes  of  the  same  star,  or  different  stars,  and  the 

elapsed  time  between  the  observations 257 

General  solution 258 

CAILLET'S  formulae  for  a  fixed  star  or  the  sun 264 

Correction  of  this  method  for  the  sun „ 266 

6th  Method. — By  two  altitudes  of  the  same  or   different   star!,   with  the 

differenceof  their  azimuths 277 

7th  Method. — By  two  different  stars  observed  at  the  same  altitude,  when 

the  time  is  given 277 

At  nearly  the  same  altitude,  observed  with  the  zenith  telescope 279 

8th  Method. — By  three  stars  observed  at  the  same  altitude  (GAUSS'S  method)  280 

The  same  by  CAGNOLI'S  formulae 286 

By  a  number  of  stars  observed  at  the  same  altitude,  treated  by  the 

Method  of  Least  Squares 289 

9th  Method. — By  tlie  transits  of  stars    over  vertical    circles  (see  Vol.   II., 

Transit  Instrument  in  the  Prime  Vertical) 293 


CONTENTS. 

PAGB 

10th  Method. — By  altitudes  near  the  meridian  when  the  time  is  not  known...  296 
(A.)  By  two  altitudes  near  the  meridian  and  the  chronometer  times  of 
the  observations,  when  the  rate  of  the  chronometer  is  known,  but  not 

its  correction 296 

(B.)  By  three  altitudes  near  the  meridian  and  the  chronometer  times 
of  the  observations,  when  neither  the  correction  nor  the  rate  of  the 

chronometer  is  known 299 

(C.)  By   two   altitudes    near  the  meridian   and  the  difference  of  the 

azimuths 301 

(D.)  By    three    altitudes   near   the    meridian   and   the   differences   of 

azimuths 302 

llth  Method. — By  the  rate  of  change  of  altitude  near  the  prime  vertical 303 

FINDING  THE  LATITUDE  AT  SEA 304 

1st  Method. — By  meridian  altitudes ,. .4 304 

2d  Method. — By  reduction  to  the  meridian  when  the  time  is  given 307 

3d   Method. — By  two  altitudes  near  the   meridian  when  the  time  is  not 

known 307 

4th  Method. — By  three  altitudes  near  the  meridian  when  the  time  is  not 

known 309 

5th  Method. — By  a  single  altitude  at  a  given  time 310 

6th  Method. — By  the  change  of  altitude  near  the  prime  vertical oil 

7th  Method.— By  the  Pole  Star 311 

8th  Method. — By  two  altitudes  with  the  elapsed  time  between  them 313 

CHAPTER  VII. 

FINDING  THE  LONGITUDE  BY  ASTRONOMICAL  OBSERVATIONS 317 

1st  Method. — By  portable  chronometers 317 

Chronometric  expeditions 323 

2d  Method.— By  signals 337 

Terrestrial  signals 337 

Celestial  signals,  (a)  Bursting  of  a  meteor,  (ft)  Beginning  or  end- 
ing of  an  eclipse  of  the  moon,  (c)  Eclipses  of  Jupiter's  satellites. 
(d)  Occupations  of  Jupiter's  satellites,  (e)  Transits  of  the  satel- 
lites over  Jupiter's  disc.  (/)  Transits  of  the  shadows  of  the  satel- 
lites over  Jupiter's  disc,  (g]  Eclipses  of  the  sun,  Occultations  of 

stars  and  planets  by  the  moon.      [See  Chapter  X.] 339 

3d  Method. — By  the  electric  telegraph 341 

Method  of  star  signals 342 

4th  Method. — By  moon  culminations 350 

PEIRCE'S  method  of  correcting  the  ephemeris 358 

Combination  of  moon  culminations  by  weights 363 

5th  Method. — By  azimuths  of  the  moon,  or  transits  of  the  moon  and  a  star 

over  the  same  vertical  circle 371 

6thMethod. — By  altitudes  of  the  moon 382 

(A.) — By  the  moon's  absolute  altitude 383 

(B.) — By  equal  altitudes  of  the  moon  and  a  star  observed  with  the 

Zenith  Telescope 386 

7th  Method. — By  lunar  distances 393 

(A.)— Rigorous  method 395 

(B.) — Approximative  method 402 


CONTENTS. 

PAGE 

FINDING  THE  LONGITUDE  AT  SEA 420 

By  chronometers 420 

By  lunar  distances 422 

By  the  eclipses  of  Jupiter's  satellites 423 

By  the  moon's  altitude 423 

By  occultations  of  stars  by  the  moon 424 

CHAPTER  VIII. 
FINDING  A  SHIP'S  PLACE  AT  SEA  BY  CIRCLES  or  POSITION — SUMNER'S  METHOD...  424 

CHAPTER  IX. 
THE  MERIDIAN  LINE  AND  VARIATION  or  THE  COMPASS 429 

CHAPTER  X. 

ECLIPSES 436 

Solar  Eclipses.    Prediction  for  the  earth  generally 436 

Fundamental  equations 439 

Outline  of  the  shadow 456 

Rising  and  setting  limits 466 

Curve  of  maximum  in  horizon , 475 

Northern  and  southern  limits 480 

Curve  of  central  eclipse 491 

Limits  of  total  or  annular  eclipse 498 

Prediction  for  a  given  place 505 

Correction  for  atmospheric  refraction  in  eclipses 515 

Correction  for  the  height  of  the  observer  above  the  level  of  the  sea 517 

Application  of  observed  solar  eclipses  to  the  determination  of  terrestrial  longi- 
tudes and  the  correction  of  the  elements  of  the  computation 518 

Lunar  eclipses 542 

Occultations  of  fixed  stars  by  the  moon 549 

Terrestrial  longitudes  from  occultations  of  stars 550 

Prediction  of  occultations 557 

Limiting  parallels 561 

Occultations  of  planets  by  the  moon 565 

Apparent  form  of  a  planet's  disc 566 

Terrestrial  longitude  from  occultations  of  planets 578 

Transits  of  Venus  and  Mercury 591 

Determination  of  the  solar  parallax t 592 

Prediction  for  the  earth  generally 593 

Occultation  of  a  fixed  star  by  a  planet 601 

CHAPTER  XL 
PRECESSION,  NUTATION,  ABERRATION,  AND  ANNUAL  PARALLAX   OF   THE  FIXED 

STARS 602 

Precession 604 

Nutation 624 

Aberration 628 

Parallax 643 

Mean  and  apparent  places  of  stars 645 


CONTENTS. 
CHAPTER  XII. 

PAGE 

DETERMINATION  OP  THE  OBLIQUITY  OF  THE  ECLIPTIC  AND  THE  ABSOLUTE  RIGHT 

ASCENSIONS  AND  DECLINATIONS  OF  STARS  BY  OBSERVATION 658 

Obliquity  of  the  ecliptic 659 

Equinoctial  points,  and  absolute  right  ascension  and  declination  of  the  fixed 

stars rr 665 

CHAPTER  XIII. 

DETERMINATION  OF  ASTRONOMICAL  CONSTANTS  BY  OBSERVATION 671 

Constants  of  refraction 671 

Constant  of  solar  parallax 673 

Constant  of  lunar  parallax 680 

Mean  semidiameters  of  the  planets 687 

Constant  of  aberration  and  heliocentric  parallax  of  fixed  stars 688 

Constant  of  nutation 698 

Constant  of  precession 701 

Motion  of  the  sun  in  space 703 


SPHEEICAL  ASTRONOMY. 


CHAPTER  I. 

THE    CELESTIAL    SPHERE SPHEEICAL    AND    RECTANGULAR 

CO-ORDINATES. 

1.  FROM  whatever  point  of  space  an  observer  be  supposed  to 
view  the  heavenly  bodies,  they  will  appear  to  him  as  if  situated 
upon  the  surface  of  a  sphere  of  which  his  eye  is  the  centre.  If, 
without  changing  his  position,  he  directs  his  eye  successively  to 
the  several  bodies,  he  may  learn  their  relative  directions,  but 
cannot  determine  either  their  distances  from  himself  or  from 
each  other. 

The  position  of  an  observer  on  the  surface  of  the  earth  is, 
however,  constantly  changing,  in  consequence,  1st,  of  the  diur> 
nal  motion,  or  the  rotation  of  the  earth  on  its  axis ;  2d,  of  the 
annual  motion,  or  the  motion  of  the  earth  in  its  orbit  around 
the  sun. 

The  changes  produced  by  the  diurnal  motion,  in  the  appa- 
rent relative  positions  or  directions  of  the  heavenly  bodies,  are 
different  for  observers  on  different  parts  of  the  earth's  surface, 
and  can  be  subjected  to  computation  only  by  introducing  the 
elements  of  the  observer's  position,  such  as  his  latitude  and 
longitude. 

But  the  changes  resulting  from  the  annual  motion  of  the 
earth,  as  well  as  from  the  proper  motions  of  the  celestial  bodies 
themselves,  may  be  separately  considered,  and  the  directions 
of  all  the  known  celestial  bodies,  as  they  would  be  seen  from 
the  centre  of  the  earth  at  any  given  time,  may  be  computed 

VOL.  I.— 2  17 


18  THE    CELESTIAL    SPHERE. 

according  to  the  laws  which  have  been  found  to  govern  the 
motions  of  these  bodies,  from  data  furnished  by  long  series  of 
observations.  The  complete  investigation  of  these  changes  and 
their  laws  belongs  to  Physical  Astronomy,  and  requires  the  consi- 
deration of  the  distances  and  magnitudes  as  well  as  of  the  direc- 
tions of  the  bodies  composing  the  system. 

Spherical  Astronomy  treats  specially  of  the  directions  of  the 
heavenly  bodies ;  and  in  this  branch,  therefore,  these  bodies  are 
at  any  given  instant  regarded  as  situated  upon  the  surface  of  a 
sphere  of  an  indefinite  radius  described  about  an  assumed 
centre.  It  embraces,  therefore,  not  only  the  problems  which  arise 
from  the  diurnal  motion,  but  also  such  as  arise  from  the  annual 
motion  so  far  as  this  affects  the  apparent  positions  of  the  hea- 
venly bodies  upon  the  celestial  sphere,  or  their  directions  from 
the  assumed  centre. 


SPHERICAL   CO-ORDINATES. 

2.  The  direction  of  a  point  may  be  expressed  by  the  angles 
which  a  line  drawn  to  it  from  the  centre  of  the  sphere,  or  point 
of  observation,  makes  with  certain  fixed  lines  of  reference.    But, 
since  such  angles  are  directly  measured  by  arcs  on  the  surface 
of  the  sphere,  the  simplest  method  is  to  assign  the  position  in 
which  the  point  appears  when  projected  upon  the  surface  of  the 
sphere.    For  this  purpose,  a  great  circle  of  the  sphere,  supposed 
to  be  given  in  position,  is  assumed  as  a  primitive  circle  of  refer- 
ence, and  all  points  of  the  surface  are  referred  to  this  circle  by  a 
system  of  secondaries  or  great  circles  perpendicular  to  the  primi- 
tive and,  consequently,  passing  through  its  poles.     The  position 
of  a  point  on  the  surface  will  then  be  expressed  by  two  spherical 
co-ordinates:  namely,  1st,  the  distance  of  the  point  from  the  pri- 
mitive circle,  measured  on  a  secondary;    2d,  the  distance  inter- 
cepted on  the  primitive  between  this  secondary  and  some  given 
point  of  the  primitive  assumed  as  the  origin  of  co-ordinates. 

We  shall  have  different  systems  of  co-ordinates,  according  to 
the  circle  adopted  as  a  primitive  circle  and  the  point  assumed  as 
the  origin. 

3.  First  system  of  co-ordinates. — Altitude  and  azimuth.— In.  this 
system,  the  primitive  circle  is  the  horizon,  which  is  that  great 
circle  of  the    sphere  whose  plane    touches  the  surface  of  the 


SPHERICAI     CO-ORDIXATES.  19 

earth  at  the  observer.*  The  plane  of  the  horizon  may  be  con- 
ceived as  that  which  sensibly  coincides  with  the  surface  of  a 
fluid  at  rest. 

The  vertical  line  is  a  straight  line  perpendicular  to  the  plane 
of  the  horizon  at  the  observer.  It  coincides  with  the  direction 
of  the  plumb  line,  or  the  simple  pendulum  at  rest.  The  two 
points  in  which  this  line,  infinitely  produced,  meets  the  sphere, 
are  the  zenith  and  nadir,  the  first  above,  the  second  below  the 
horizon. 

The  zenith  and  nadir  are  the  poles  of  the  horizon. 

Secondaries  to  the  horizon  are  vertical  circles.  They  all  pass 
through  the  zenith  and  nadir,  and  their  planes,  which  are  called 
vertical  planes,  intersect  in  the  vertical  line. 

Small  circles  parallel  to  the  horizon  are  called  almucantars,  or 
parallels  of  altitude. 

The  celestial  meridian  is  that  vertical  circle  whose  plane  passes 
through  the  axis  of  the  earth  and,  consequently,  coincides  with 
the  plane  of  the  terrestrial  meridian.  The  'intersection  of  this 
plane  with  the  plane  of  the  horizon  is  the  meridian  line,  and  the 
points  in  which  this  line  meets  the  sphere  are  the  north  and  south 
points  of  the  horizon,  being  respectively  north  and  south  of  the 
plane  of  the  equator. 

The  prime  vertical  is  the  vertical  circle  which  is  perpendicular 
to  the  meridian.  The  line  in  which  its  plane  intersects  the 
plane  of  the  horizon  is  the  east  and,  west  line,  and  the  points  in 
which  this  line  meets  the  sphere  are  the  east  and  west  points  of 
the  horizon. 

The  north  and  south  points  of  the  horizon  are  the  poles  of  the 
prime  vertical,  and  the  east  and  west  points  are  the  poles  of  the 
meridian.  * 


*  In  this  definition  of  the  horizon  we  consider  the  plane  tangent  to  the  earth's 
surface  as  sensibly  coinciding  with  a  parallel  plane  passed  through  the  centre  ;  that 
is,  we  consider  the  radius  of  the  celestial  sphere  to  be  infinite,  and  the  radius  of  the 
earth  to  be  relatively  zero.  In  general,  any  number  of  parallel  planes  at  finite  dis- 
tances must  be  regarded  as  marking  out  upon  the  infinite  sphere  the  same  great  circle. 
Indeed,  since  in  the  celestial  sphere  we  consider  only  direction,  abstracted  from  dis- 
tance, all  lines  or  planes  having  the  same  direction — that  is,  all  parallel  lines  or 
planes — must  be  regarded  as  intersecting  the  surface  of  the  sphere  in  the  same 
point  or  the  same  great  circle.  The  point  of  the  surface  of  the  sphere  in  which  a 
number  of  parallel  lines  are  conceived  to  meet  is  called  the  vanishing  point  of  those 
lines;  and,  in  like  manner,  the  great  circle  in  which  a  number  of  parallel  planes  are 
conceived  to  meet  may  be  called  the  vanishing  circle  of  those  planes. 


20  THE    CELESTIAL    SPHERE. 

The  altitude  of  a  point  of  the  celestial  sphere  is  its  distance 
from  the  horizon  measured  on  a  vertical  circle,  and  its  azimuth  is 
the  arc  of  the  horizon  intercepted  between  this  vertical  circle 
and  any  point  of  the  horizon  assumed  as  an  origin.  The  origin 
from  which  azimuths  are  reckoned  is  arbitrary ;  so  also  is  the 
direction  in  which  they  are  reckoned ;  but  astronomers  usually 
take  the  south  point  of  the  horizon  as  the  origin,  and  reckon 
towards  the  right  hand,  from  0°  to  360°  ;  that  is,  completely 
around  the  horizon  in  the  direction  expressed  by  writing  the 
cardinal  points  of  the  horizon  in  the  order  S.W.  X.  E.  We 
may,  therefore,  also  define  azimuth  as  the  angle  which  the 
vertical  plane  makes  with  the  plane  of  the  meridian. 

Navigators,  however,  usually  reckon  the  azimuth  from  the 
north  or  south  points,  according  as  they  are  in  north  or  south 
latitude,  and  towards  the  east  or  west,  according  as  the  point 
of  the  sphere  considered  is  east  or  west  of  the  meridian:  so  that 
the  azimuth  never  exceeds  180°.  Thus,  an  azimuth  which  is 
expressed  according  to  the  first  method  simply  by  200°  would 
be  expressed  by  a  navigator  in  north  latitude  by  N.  20°  E.,  and 
by  a  navigator  in  south  latitude  by  S.  160°  E.,  the  letter  prefixed 
denoting  the  origin,  and  the  letter  affixed  denoting  the  direction 
in  which  the  azimuth  is  reckoned,  or  whether  the  point  consi- 
dered is  east  or  west  of  the  meridian. 

When  the  point  considered  is  in  the  horizon,  it  is  often 
referred  to  the  east  or  west  points,  and  its  distance  from  the 
nearest  of  these  points  is  called  its  amplitude.  Thus,  a  point  in 
the  horizon  whose  azimuth  is  110°  is  said  to  have  an  amplitude 
of  W.  20°  K 

Since  by  the  diurnal  motion  the  observer's  horizon  is  made 
to  change  its  position  in  the  heavens,  the  co-ordinates,  altitude 
and  azimuth,  are  continually  changing.  Their  values,  therefore, 
will  depend  not  only  upon  the  observer's  position  on  the  earth, 
but  upon  the  time  reckoned  at  his  meridian. 

Instead  of  the  altitude  of  a  point,  we  frequently  employ  its 
zenith  distance,  which  is  the  arc  of  the  vertical  circle  between  the 
point  and  the  zenith.  The  altitude  and  zenith  distance  are, 
therefore,  complements  of  each  other. 

We  shall  hereafter  denote  altitude  by- A,  zenith  distance  by  £, 
azimuth  by  A.  We  shall  have  then 

C  =  90°  —  h  h  =  90°  —  C 


SPHERICAL    CO-ORDINATES.  21 

The  value  of  £  for  a  point  below  the  horizon  will  be  greater 
than  90°,  and  the  corresponding  value  of  h,  found  by  the  for- 
mula h  =  90°  —  £,  will  be  negative  :  so  that  a  negative  altitude 
will  express  the  depression  of  a  point  below  the  horizon.  Thus, 
a  depression  of  10°  will  be  expressed  by  h  =  —10°,  or  £  —  100°. 

4.  Second  system  of  co-ordinates. — Declination  and  hour  angle. — In 
this  system,  the  primitive  circle  is  the  celestial  equator,  or  that 
great  circle  of  the  sphere  whose  plane  is  perpendicular  to  the 
axis  of  the  earth  and,  consequently^  coincides  with  the  plane  of 
the  terrestrial  equator.  This  circle  is  also  sometimes  called  the 
Equinoctial. 

The  diurnal  motion  of  the  earth  does  not  change  the  position 
of  the  plane  of  the  equator.  The  axis  of  the  earth  produced  to 
the  celestial  sphere  is  called  the  axis  of  the  heavens:  the  points 
in  which  it  meets  the  sphere  are  the  north  and  south  poles*of 
the  equator,  or  the  poles  of  the  heavens. 

Secondaries  to  the  equator  are  called  circles  of  declination',  and 
also  hour  circles.  Since  the  plane  of  the  celestial  meridian 
passes  through  the  axis  of  the  equator,  it  is  also  a  secondary  to 
the  equator,  and  therefore  also  a  circle  of  declination. 

Parallels  of  declination  are  small  circles  parallel  to  the  equator. 

The  declination  of  a  point  of  the  sphere  is  its  distance  from  the 
equator  measured  on  a  circle  of  declination,  and  its  hour  angle  is 
the  angle  at  either  pole  between  this  circle  of  declination  and  the 
meridian.  The  hour  angle  is  measured  by  the  arc  of  the  equator 
intercepted  between  the  circle  of  declination  and  the  meridian. 
As  the  meridian  and  equator  intersect  in  two  points,  it  is  neces- 
sary to  distinguish  which  of  these  points  is  taken  as  the  origin 
of  hour  angles,  and  also  to  know  in  what  direction  the  arc  which 
measures  the  hour  angle  is  reckoned.  Astronomers  reckon 
from  that  point  of  the  equator  which  is  on  the  meridian  above 
the  horizon,  towards  the  west, — that  is,  in  the  direction  of  the 
apparent  diurnal  motion  of  the  celestial  sphere, — and  from  0°  to 
360°,  or  from  Qh  to  24*,  allowing  15°  to  each  hour. 

Of  these  co-ordinates,  the  declination  is  not  changed  by  the 
diurnal  motion,  while  the  hour  angle  depends  only  on  the  time 
at  the  meridian  of  the  observer,  or  (which  is  the  same  thing)  on 
the  position  of  his  ^meridian  in  the  celestial  sphere.  All  the 
observers  on  the  same  meridian  at  the  same  instant  will,  for  the 
same  star,  reckon  the  same  declination  and  hour  angle.  We  have 


22  THE    CELESTIAL    SPHERE. 

thus  introduced  co-ordinates  of  which  one  is  wholly  independent 
of  the  observer's  position  and  the  other  is  independent  of  his 
latitude. 

We  shall  denote  declination  by  o,  and  north  declination  will 
be  distinguished  by  prefixing  to  its  numerical  value  the  sign  -f-, 
and  south  declination  by  the  sign  — . 

We  shall  sometimes  make  use  of  the  polar  distance  of  a  point, 
or  its  distance  from  one  of  the  poles  of  the  equator.  If  we  denote 
it  by  P,  the  north  polar  distance  will  be  found  by  the  formula 

Pr=90°    —  d 

and  the  south  polar  distance  by  the  formula 

P  =  90°  +  d 

The  hour  angle  will  generally  be  denoted  by  t.  It  is  to  be 
ol^erved  that  as  the  hour  angle  of  a  celestial  body  is  continually 
increasing  in  consequence  of  the  diurnal  motion,  it  may  be  con- 
ceived as  having  values  greater  than  360°,  or  24*,  or  greater  than 
any  given  multiple  of  360°.  Such  an  hour  angle  may  be  re- 
garded as  expressing  the  time  elapsed  since  some  given  passage 
of  the  body  over  the  meridian.  But  it  is  usual,  when  values 
greater  than  360°  result  from  any  calculation,  to  deduct  360°. 
Again,  since  hour  angles  reckoned  towards  the  west  are  always 
positive,  hour-angles  reckoned  towards  the  east  must  have  the 
negative  sign :  so  that  an  hour  angle  of  300°,  or  20A,  may  also  be 
expressed  by  — 60°,  or  — 47'. 

5.  Third  system  of  co-ordinates., — Declination  and  right  ascension. — 
In  this  system,  the  primitive  plane  is  still  the  equator,  and  the 
first  co-ordinate  is  the  same  as  in  the  second  system,  namely,  the 
declination.  The  second  co-ordinate  is  also  measured  on  the 
equator,  but  from  an  origin  which  is  not  affected  by  the  diurnal 
motion.  Any  point  of  the  celestial  equator  might  be  assumed 
as  the  origin;  but  that  which  is  most  naturally  indicated  is 
the  vernal  equinox,  to  define  which  some  preliminaries  are 
necessary. 

The  ecliptic  is  the  great  circle  of  the  celestial  sphere  in  which 
the  sun  appears  to  move  in  consequence  of  the  earth's  motion  in 
its  orbit.  The  position  of  the  ecliptic  is  not  absolutely  fixed  in 
space  ;  but,  according  to  the  definition  just  given,  its  position  at 

iy  instant  coincides  with  that  of  the  great  circle  in  which  the 


SPHERICAL    CO-ORDINATES.  23 

sun  appears  to  be  moving  at  that  instant.  Its  annual  change  is, 
however,  very  small,  and  its  daily  change  altogether  insensible. 

The  obliquity  of  the  ecliptic  is  the  angle  which  it  makes  with 
the  equator. 

The  points  where  the  ecliptic  and  equator  intersect  are  called 
the  equinoctial  points,  or  the  equinoxes  ;  and  that  diameter  of  the 
sphere  in  which  their  planes  intersect  is  the  line  of  equinoxes. 

Tl^e  vernal  equinox  is  the  point  through  which  the  sun  ascends 
from  the  southern  to  the  northern  side  of  the  equator ;  and  the 
autumnal  equinox  is  that  through  which  the  sun  descends  from  the 
northern  to  the  southern  side  of  the  equator. 

The  solstitial  points,  or  solstices,  are  the  points  of  the  ecliptic 
90°  from  the  equinoxes.  They  are  distinguished  as  the  north- 
ern and  southern,  or  the  summer  and  winter  solstices. 

The  equinoctial  colure  is  the  circle  of  declination  which  passes 
through  the  equinoxes.  The  solstitial  colure  is  the  circle  of  decli- 
nation which  passes  through  the  solstices.  The  equinoxes  are 
the  poles  of  the  solstitial  colure. 

By  the  annual  motion  of  the  earth,  its  axis  is  carried  very 
nearly  parallel  to  itself,  so  that  the  plane  of  the  equator,  which 
is  always  at  right  angles  to  the  axis,  is  very  nearly  a  fixed  plane 
of  the  celestial  sphere.  The  axis  is,  however,  subject  to  small 
changes  of  direction,  thg  effect  of  which  is  to  change  the 
position  of  the  intersection  of  the  equator  and  the  ecliptic,  and 
hence,  also,  the  position  of  the  equinoxes.  In  expressing  the 
positions  of  stars,  referred  to  the  vernal  equinox,  at  any  given 
instant,  the  actual  position  of  the  equinox  at  the  instant  is 
understood,  unless  otherwise  stated. 

The  right  ascension  of  a  point  of  the  sphere  is  the  arc  of  the 
equator  intercepted  between  its  circle  of  declination  and  the 
vernal  equinox,  and  is  reckoned  from  the  vernal  equinox  east- 
ward from  0°  to  360°,  or,  in  time,  from  0*  to  247'. 

The  point  of  observation  being  supposed  at  the  centre  of  the 
earth,  neither  the  declination  nor  the  right  ascension  will  be 
affected  by  the  diurnal  motion:  so  that  these  co-ordinates  are 
wholly  independent  of  the  observer's  position  on  the  surface  of 
the  earth.  Their  values,  therefore,  vary  only  with  the  time, 
and  are  given  in  the  ephemerides  as  functions  of  the  time 
reckoned  at  some  assumed  meridian. 

We  shall  generally  denote  right  ascension  by  a.  As  its  value 
reckoned  towards  the  east  is  positive,  a  negative  value  resulting 


24  THE    CELESTIAL    SPHERE. 

from  any  calculation  would  be  interpreted  as  signifying  an  arc 
of  the  equator  reckoned  from  the  vernal  equinox  towards  the 
west.  Thus,  a  point  whose  right  ascension  is  300°,  or  20*,  may 
also  be  regarded  as  in  right  ascension  — 60°,  or  — 4/l ;  but  such 
negative  values  are  generally  avoided  by  adding  360°,  or  24*. 
Again,  in  continuing  to  reckon  eastward  we  may  arrive  at 
values  of  the  right  ascension  greater  than  24*,  or  greater  than 
48*,  etc.;  but  in  such  cases  we  have  only  to  reject  24*,  48*,  etc. 
to  obtain  values  which  express  the  same  thing. 

6.  Fourth   system   of  co-ordinates. —  Celestial    latitude  and    longi- 
tude.— In  this  system  the  ecliptic  is  taken  as  the  primitive  circle, 
and  the  secondaries  by  which  points  of  the  sphere  are  referred 
to  it  are  called  circles  of  latitude.     Parallels  of  latitude  are  small 
circles  parallel  to  the  ecliptic. 

The  latitude  of  a  point  of  the  sphere  is  its  distance  from  the 
ecliptic  measured  on  a  circle  of  latitude,  and  its  longitude  is  the 
arc  of  the  ecliptic  intercepted  between  this  circle  of  latitude  and 
the  vernal  equinox.  The  longitude  is  reckoned  eastward  from 
0°  to  360°.  The  longitude  is  sometimes  expressed  in  signs, 
degrees,  &c.,  a  sign  being  equal  to  30°,  or  one-twelfth  of  the 
ecliptic. 

These  co-ordinates  are  also  independent  of  the  diurnal  motion. 
It  is  evident,  however,  that  the  system  of  declination  and  right 
ascension  will  be  generally  nrore  convenient,  since  it  is  more 
directly  related  to  our  first  and  second  systems,  which  involve 
the  observer's  position. 

"We  shall  denote  celestial  latitude  by  /?;  longitude  by  L  Posi- 
tive values  of  /9  belong  to  points  on  the  same  side  of  the  ecliptic 
as  the  north  pole;  negative  values,  to  those  on  the  opposite  side. 

In  connection  with  this  system  we  may  here  define  the  nona- 
gesimal,  which  is  that  point  of  the  ecliptic  which  is  at  the  greatest 
altitude  above  the  horizon  at  any  given  time.  That  vertical 
circle  of  the  observer  which  is  perpendicular  to  the  ecliptic  meets 
it  in  the  nonagesimal ;  and,  being  a  secondary  to  the  ecliptic,  it 
is  also  a  circle  of  latitude :  it  is  the  great  circle  which  passes 
through  the  observer's  zenith  and  the  pole  of  the  ecliptic. 

7.  Co-ordinates  of  the  observer  s  position. — We  have  next  to  ex- 
press the  position  of  the  observer  on  the  surface  of  the  earth, 
according  to  the  different  systems  of  co-ordinates.     This  will  be 


SPHERICAL    CO-ORDINATES.  25 

done  by  referring  his  zenith  to  the  primitive  circle  in  the  same 
manner  as  in  the  case  of  any  other  point. 

In  the  first  system,  the  primitive  circle  being  the  horizon,  of 
which  the  zenith  is  the  pole,  the  altitude  of  the  zenith  is  always 
90°,  and  its  azimuth  is  indeterminate. 

In  the  second  system,  the  declination  of  the  zenith  is  the  same 
as  the  terrestrial  latitude  of  the  observer,  and  its  hour  angle  is 
zero.  The  declination  of  the  zenith  of  a  place  is  called  the 
geographical  latitude,  or  simply  the  latitude,  and  will  be  hereafter 
denoted  by  (p.  North  latitudes  will  have  the  sign  -f ;  south 
latitudes,  the  sign  — . 

In  the  third  system,  the  declination  of  the  zenith  is,  as  before, 
the  latitude  of  the  observer,  and  its  right  ascension  is  the  same 
as  the  hour  angle  of  the  vernal  equinox. 

In  the  fourth  system,  the  celestial  latitude  of  the  zenith  is  the 
same  as  the  zenith  distance  of  the  nonagesimal,  and  its  celestial 
longitude  is  the  longitude  of  the  nonagesimal. 

It  is  evident,  from  the  definitions  which  have  been  given,  that 
the  problem  of  determining  the  latitude  of  a  place  by  astro- 
nomical observation  is  the  same  as  that  of  determining  the 
declination  of  the  zenith ;  and  the  problem  of  finding  the  lon- 
gitude may  be  resolved  into  that  of  determining  the  right 
ascension  of  the  meridian  at  a  time  when  that  of  the  prime 
meridian  is  also  given,  since  the  longitude  is  the  arc  of  the 
equator  intercepted  between  the  two  meridians,  and  is,  conse- 
quently, the  difference  of  their  right  ascensions. 

8.  The  preceding  definitions  are  exemplified  in  the  following 
figures. 

Fig.  1  is  a  stereographic  projection  of 
the  sphere  upon  the  plane  of  the  horizon, 
the  projecting  point  being  the  nadir.  Since 
the  planes  of  the  equator  and  horizon  are 
both  perpendicular  to  that  of  the  meridian, 
their  intersection  is  also  perpendicular  to 
it;  and  hence  the  equator  WQE  passes 
through  the  east  and  west  points  of  the 
horizon.  All  vertical  circles  passing 
through  the  projecting  point  will  be  projected  into  straight 
lines,  as  the  meridian  NZS,  the  prime  vertical  WZE,  and  the 
vertical  circle  ZOH  drawn  through  any  point  0  of  the  surface 


26  THE    CELESTIAL    SPHERE. 

of  the  sphere.     "We  have  then,  according  to  the  notation  adopted 
in  the  first  system  of  co-ordinates, 

h  =  the  altitude  of  the  point  0  =  OH, 
C  —  the  zenith  distance  "  =  OZ, 
A  =  the  azimuth  "  =  SH,  or 

=  the  angle  SZH. 

If  the  declination  circle  POD  be  drawn,  we  have,  in  the  second 
system  of  co-ordinates, 

d  =  the  decimation  of  0  =  OD, 
P  =  the  polar  distance  "  =  PO, 
t  =  the  hour  angle  "  =  ZPD,  or  =  QD. 

If  Vis  the  vernal  equinox,  we  have,  in  the  third  system  of 
co-ordinates, 

d  =  the  declination  of  0  =  OD, 
a  =  the  right  ascension  =  FT),  or 

=  the  angle  VPD. 

In  this  figure  is  also  drawn  the  six  hour  circle  JSPW,  or  the 
declination  circle  passing  through  the  east  and  west  points  of  the 
horizon.1  The  angle  ZP  W,  or  the  arc  Q  W,  being  90°,  the  hour 
angle  of  a  point  on  this  circle  is  either  -f-  67'  or  — 6A,  that  is,  either 
6A  or  18*. 

Fig.  2  is  a  repetition  of  the  preceding  figure,  with  the  addi- 
ng. 2.  tion  of   the  ecliptic  and  the  circles  related 
to  it.      C  VT  represents  the  ecliptic,  Pf  its 
pole,  P'OL  a  circle  of  latitude.     Hence  we 
r    have,  in  our  third  system  of  co-ordinates, 


IT. 


ft  =  the  celestial  latitude  of  0  =  OL, 
X  =  longitude  "     =E±  VL, 

=  the  angle  VP'L. 

We  have  also  FPthe  equinoctial  colure,  P'PAB  the  solstitial 
colure,  P'ZGF  the  vertical  circle  passing  through  P',  which  is 
therefore  perpendicular  to  the  ecliptic  at  G.  The  point  G  is  the 
nonagesimal ;  ZG-  is  its  zenith  distance,  VG  its  longitude ;  or 
ZG  is  the  celestial  latitude  and  VG  the  celestial  longitude  of  the 
zenith. 

Finally,  we  have,  in  both  Fig.  1  and  Fig.  2, 

(p  =  the  geographical  latitude  of  the  observer 
=  ZQ  =  90°  —  PZ  =  PN 


SPHERICAL    CO-ORDINATES.  27 

Hence  the  latitude  of  the  observer  is  always  equal  to  the  alti' 
tude  of  the  north  pole.  For  an  observer  in  south  latitude,  the 
north  pole  is  below  the  horizon,  arid  its  altitude  is  a  negative 
quantity:  so  that  the  definition  of  latitude  as  the  altitude  of  the 
north  pole  is  perfectly  general  if  we  give  south  latitudes  the 
negative-  sign.  The  south  latitude  of  an  observer  considered 
independently  of  its  sign  is  equal  to  the  altitude  of  the  south 
pole  above  his  horizon,  the  elevation  of  one  pole  being  always 
equal  to  the  depression  of  the  other. 

9.  Numerical  expression  of  hour  angles.  —  The  equator,  upon 
which  hour  angles  are  measured,  may  be  conceived  to  be  divided 
into  24  equal  parts,  each  of  which  is  the  measure  of  one  hour, 
and  is  equivalent  to  2\  of  360°,  or  to  15°.  The  hour  is  divided 
sexagesimally  into  minutes  and  seconds  of  time,  distinguished 
from  minutes  and  seconds  of  arc  by  the  letters  "l  and  "  instead 
of  the  accents  '  and  ".  We  shall  have,  then, 


To  convert  an  angle  expressed  in  time  into  its  equivalent  in 
arc,  multiply  by  15  and  change  the  denominations  h  m  *  into 
0  '  "  ;  and  to  convert  arc  into  time,  divide  by  15  and  change  °  '  " 
into  h  m  '.  The  expert  computer  will  readily  find  ways  to 
abridge  these  operations  in  practice.  It  is  well  to  observe,  for 
this  purpose,  that  from  the  above  equalities  we  also  have, 

1°  =  4-         1'  =  4s 

and  that  we  may  therefore  convert  degrees  and  minutes  of  arc 
into  time  by  multiplying  by  4  and  changing  °  '  into  m  *  ;  and 
reciprocally. 

TRANSFORMATION    OF    SPHERICAL    CO-ORDINATES.. 

10.  Given  the  altitude  (h)  and  azimuth  (A)  of  a  star,  or  of  any  point 
of  the  sphere,  and  the  latitude  (?)  of  the  observer,  to  find  the  decima- 
tion (3)  and  hour  angle  (t)  of  the  star  or  the  point.  In  other  words, 
to  transform  the  co-ordinates  of  the  first  system  into  those  of  the 
second. 

This  problem  is  solved  by  a  direct  application  of  the  formulae 
of  Spherical  Trigonometry  to  the  triangle  POZ,  Fig.  1,  in  which, 
0  being  the  given  star  or  point,  we  have  three  parts  given, 


28 


THE    CELESTIAL    SPHERE. 


Fig.  3. 


namely,  ZO  the  zenith  distance  or  complement  of  the  given 
altitude,  PZO  the  supplement  of  the  given  azimuth,  and  PZ  the 
complement  of  the  given  latitude;  from  which  we  can  find  PO 
the  polar  distance  or  complement  of  the  required  declination, 
and  ZPO  the  required  hour  angle.  But,  to  avoid  the  trouble  of 
taking  complements  and  supplements,  the  formulae  are'  adapted 
so  as  to  express  the  declination  and  hour  angle  directly  in  terms 
of  the  altitude,  azimuth,  and  latitude. 

To  show  as  clearly  as  possible  how  the  formulae 
of  Spherical  Trigonometry  are  thus  converted  into 
formulae  of  Spherical  Astronomy,  let  us  first  con- 
sider a  spherical  triangle  ABC,  Fig.  3,  in  which 
there  are  given  the  angle  A,  and  the  sides  b  and  c,  to 
find  the  angle  B  and  the  side  a.  The  general  rela- 
tions between  these  five  quantities  are  [Sph.  Trio-. 
Art.  114]* 

cos  a  =  co8  c  cos  b  -[-  sin  c'mn  b  cos  A        ^ 
sin  a  cos  B  =  sin  c  cos  b  —  cos  c  sin  b  cos  A          >     (QV) 
sin  a  sin  B  =  sin  b  sin  A         J 

Now,  comparing  the  triangle  ABC  with  the  triangle  PZO  of 
Fig.  1,  we  have 


=  1«UU—  A  a=    lJU  =  y()"  —  d 

b=    Z0  =   90°—  h  T$  =  ZPO=t 

c=    PZ=   90°—  <p 

Substituting  these  values  in  the  above  equations,  we  obtain 

sin  d  —  sin  <p  sin  h  —  cos  <p  cos  h  cos  A  (1) 

cos  d  cos"  t  =  cos  <f  sin  h  -f-  sin  <p  cos  h  cos  J.  (2) 

cos  d  sin  f  =  cos  h  sin  .4  (3) 

which  are  the  required  expressions  of  o  and  t  in  terms  of  h  and  JL 
If  the  zenith  distance  (£)  of  the  star  is  given,  the  equations 
will  be 

sin  (5  =  sin  <p  cos  C  —  cos  9?  sin  C  cos  J.  (4) 

cos  (5  cos  ?  =  cos  <p  cos  C  +  sin  ^  sin  C  cos  ^.  (5) 

cos  <5  sin  #z=  sin  ^  sin  A  (6) 

Since,  in  Spherical  Astronomy,  we  consider  arcs  and  angles 
whose  values  may  exceed  180°,  it  becomes  necessary,  in  general, 


*  The  references  to  Trigonometry  are  to  the  5th  edition  of  the  author's  "  Treatise 
on  Plane  and  Spherical  Trigonometry." 


SPHERICAL    CO-ORDINATES.  29 

to  determine  such  arcs  and  angles  by  both  the  sine  and  the 
cosine,  in  order  to  fix  the  quadrant  in  which  their  values  are  to 
be  taken.  It  has  been  shown  in  Spherical  Trigonometry  that 
when  we  consider  the  general  triangle,  or  that  in  which  values 
are  admitted  greater  than  180°,  there  are  two  solutions  of  the 
triangle  in  every  case,  but  that  the  ambiguity  is  removed  and 
one  of  these  solutions  excluded  "  when,  in  addition  to  the  other 
data,  the  sign  of  the  sine  or  cosine  of  one  of  the  required  parts  is 
given."  [Sph.  Trig.  Art.  113.]  In  our  present  problem  the  sign 
of  cos  3  is  given,  since  it  is  necessarily  positive ;  for  d  is  always 
numerically  less  than  90°,  that  is,  between  the  limits  +90°  and 
—90°.  Hence  cos  /  has  the  sign  of  the  second  member  of  (2)  or 
(5),  and  sin  t  the  sign  of  the  second  member  of  (3)  or  (6),  and  t 
is  to  be  taken  in  the  quadrant  required  by  these  two  signs.  Since 
h  also  fells  between  the  limits  +90°  and  —90°,  or  £  between  0° 
and  180°,  cos  A,  or  sin  £,  is  positive,  and  therefore  by  (3)  or  (6) 
sin  t  has  the  sign  of  sin  A ;  that  is,  when  A  <  180°  we  have  t  < 
180°,  and  when  A>180°  we  have  *>180°, — conditions  which 
also  follow  directly  from  the  nature  of  our  problem,  since  the 
star  is  west  or  east  of  the  meridian  according  as  A  <  180°  or  A 
>  180°.  The  formula  (1)  or  (4)  fully  determines  o,  which  will 
always  be  taken  less  than  90°,  positive  or  negative  according  to 
the  sign  of  its  sine.* 

To  adapt  the  equations  (4),  (5),  and  (6)  for  logarithmic  compu- 
tation, let  m  and  M  be  assumed  to  satisfy  the  conditions  [PL 
Trig.  Art.  174], 

m  sin  M=  sin  C  cos  A  \ 

m  cos  Jf  =  cos  C  / 

the  three  equations  may  then  be  written  as  follows : 

sin  8  —  m  sin  (<p  —  M)  ^ 

cos  d  cos  t  =  m  cos  (y>  —  Jf)  v     (8) 

cos  d  sin  t  =  sin  C  sin  A  ) 

If  we  eliminate  m  from  these  equations,  the  solution  takes  the 
following  convenient  form : 


*  There  are,  however,  special  problems  in  which  it  is  convenient  to  depart  from 
this  general  method,  and  to  admit  declinations  greater  than  90°,  as  will  be  seen 
hereafter. 


30  THE    CELESTIAL    SPHERE. 

tan  J/"=tan  C  cos  A 

tan  A  sin  M 

tan  r    =—  /q\ 

COS     (y   _   J/) 

tan  3    =  tan  (<?  —  JLT)  cos  £ 

in  the  use  of  which,  we  must  observe  to  take  t  greater  or  less 
than  180°  according  as  A  is  greater  or  less  than  180°,  since  the 
hour  angle  and  the  azimuth  must  fall  on  the  same  side  of  the 
meridian. 

EXAMPLE. — In  the  latitude  <p  =  38°  58'  53",  there  are  given  for 
a  certain  star  £  =  69°  42'  30",  A  =  300°  10'  30"  ;  required  3  and  t. 
The  computation  by  (9)  may  be  arranged  as  follows:* 

log  tan  C  0.4320966 

tj)=       38°  58'  53"      log  cos  A  9.7012595  log  tan  A  w.0.2355026 

M=       53    39    41.98    log  tan  M  0.1333561  log  sin  M  9.9060828. 

$  —  M=  — 14    40   48.98     log  tan  (0  —  M)  w9.4182633  log  sec  (0  —  M)     0.0144141 

t=     304    55    26.49    log  cos  t  9.7577677  log  tan  t  nO.  1559995 

6=    —8    31    46.56    log  tan  J  n9.1760310 

Converting  the  hour  angle  into  time,  we  have  t  =  20*  19m  41".7G6. 

11.  The  angle  POZ,  Fig.  1,  between  the  vertical  circle  and 
the  declination  circle  of  a  star,  is  frequently  called  the  parattactie 
angle,  and  will  here  be  denoted  by  q.  To  find  its  value  from  the 
data  f ,  A,  and  ^,  we  have  the  equations 

cos  d  cos  q  —  sin  C  sin  y  -j-  cos  C  cos  <p  cos  A          ~\ 
cos  d  sin  q  =  cos  </>  sin  A          ) 

which  may  be  solved  in  the  following  form : 

f  sin  F=  sin  C 

/  cos  F=.  cos  C  cos  A 
cos  d  cos  q  =f  cos  (<p  —  F) 
cos  5  sin  q  =  cos.  ^  sin  A 


38  6  sin  q  =  cos.  ^  sin  A 

* 
or  in  the  following  : 

g  sin  £r  =  sin  y  ' 

tf    COS    f7  zrr  COS   C?   COS    A 

(12) 


<7  sn       =  sn  $? 

gr  COS    Gr  =  COS  fp  COS  A 

cos  5  cos  q  =  g  cos  (C  —  G-) 
cos  J  sin  #  =  cos  <p  sin  J. 

or  again  in  the  following : 

*  In  this  work  the  letter  n  prefixed  to  a  logarithm  indicates  that  the  number  to 
which  it  corresponds  is  to  have  the  negative  sign. 


SPHERICAL    CO-ORDINATES.  31 

tan  6r  =  tan  <p  sec  A 

tan  A  cos  G- 
tan   q  =  — 

cos  (C  —  6r) 

and,  in  the  use  of  the  last  form,  it  is  to  be  observed  that  q  is  to 
be  taken  greater  or  less  than  180°  according  as  A  is  greater  or 
less  than  180°,  as  is  evident  from  the  preceding  forms. 

12.  If,  in  a  given  latitude,  the  azimuth  of  a  star  of  known 
declination  is  given,  its  hour  angle  and  zenith  distance  may  be 
found  as  follows.  We  have 

cos  t  sin  tf>  —  sin  t  cot  A  —  cos  <p  tan  d 
cos  C  sin  <p  —  sin  C  cos  <f  cos  A  =  sin  d 
The  solution  of  the  first  of  these  is  effected  by  the  equations 

b  sin  B  =  sin  <p 
b  cos  B  =  cot  A 


•         ,T> 

sin  (B  — 


and  that  of  the  second  by 

c  sin  C=  sin  <p 

c  cos  C=  cos  <p  cos  A 

.  sin  d 

sin  (C— C)  =  — 
c 

13.  Finally,  if  from  the  given  altitude  and  azimuth  we  wish  to 
find  the   declination,  hour  angle,  and  parallactic  angle  at  the 
same  time,  it  will  be  convenient  to  use  Gauss's  Equations,  which 
for  the  triangle  ABC,  Fig.  3,  are 

cos  I  a  sin  J  (B  -j-  C  )  —  cos  I  (b  —  c)  cos  I  A 
cos  £  a  cos  £  (B  -f  C  )  —  cos  \  (b  -j-  c)  sin  £  A 
sin  |  a  sin  £  (B  —  C  )  —  sin  i  (6  —  c)  cos  J  A 
sin  |  a  cos  £  (B  —  C  )  =  sin  \  (b  -f  c)  sin  |  A 

which  are  to  be  solved  in  the  usual  manner  [Sph.  Trig.  Art. 
116]   after   substituting  the   values  A  =  180°  —  A,  b  =  C,  c  = 

14.  Given  the  decimation  (d)  and  hour  angle  (t)  of  a  star,  and  the 
latitude  (<p),  to  find  the  zenith  distance  (f )  and  azimuth  (A)  of  the  star. 
That  is,  to  transform  the  co-ordinates  of  the  second  system  into 
those  of  the  first. 

We  take  the  same  general  equations  (&)  of  Spherical  Trigo- 
nometry which  have  been  employed  in  the  solution  of  the  pre- 


(B) 


32 


THE    CELESTIAL    SPHERE. 


ceding  problem,  Art,  10;  but  we  now  suppose  the  letters  A,  B, 
C,  in  Fig.  3,  to  represent  respectively  the  pole,  the  zenith,  and 
the  star,  so  that  we  substitute 


and  the  equations  become 

cos  C  =  sin  <p  sin  d  -}-  cos  <p  cos  8  cos  t 
sin  C  cos  A  =  —  cos  <p  sin  d  -j-  sin  <p  cos  5  cos  £ 
sin  C  sin  A  =  cos  <5  sin  £ 

which  express  £  and  ^1  directly  in  terms  of  the  data. 
Adapting  these  for  logarithmic  computation,  we  have 

m  sin  3f=sin  «5 

m  cos  Jf  =  cos  d  cos  f 

cos  C  =  m  cos  (<p  —  Jf) 
sin  C  cos  ^4  =  m  sin  (9?  —  Jtf) 
sin  C  sin  ^4.  =  cos  <5  sin  ^ 

in  which  m  is  a  positive  number. 

Eliminating  m,  we  deduce  the  following  simple  and  accurate 
formulae : 

tan<S 


tan  A  = 


cos  t 

tan  t  cos  M 


sin  (^  —  31) 
tan  C  =  to"fr—  M) 


COS 


(10) 


where  ^4.  is  to  be  taken  greater  or  less  than  180°  according  as  t 
is  greater  or  less  than  180°. 

EXAMPLE  1.— In  latitude  <p  =  38°  58'  53",  there  are  given  for 
a  certain  star,  £  =  —  8°  31r  46".56,  /  =  20A  19M  41-.766 ;  required 
A  and  £.  By  (16)  we  have  : 

log  tan  d  nO.  1760310 

0—       38°  58' 53"      log  cos  t  9.7577677  log  tan  t  nO.  1559995 

M——  14  40   48.98    log  tan  J/  nO. 41 82633  log  cos  M  9.9855859 

6  —  31—       53   39   41.98     log  tan  (0  —  Jf)    0.1333561  log  cosec  (0  — 3/)  0.0939172 

A—     300   10   30         log  cos  A  9.7012595  log  tan  A  nO.2355026 

C  =  '     69   42   30         log  tan  C  0.4320966 


SPHERICAL    CO-ORDINATES.  33 

For  verification  we  can  use  the  equation 

sin  C  sin  A  =  cos  d  sin  t 

log  sin  C     9.9721748            log  cos  d  9.9951697 

log  sin  A    9.9367621             log  sin  t  9.9137672 

9.9089369  9.9089369 

EXAMPLE  2. — In  latitude  <p  —  — 48°  32',  there  are  given  for  a 
star,  £  =  44°  6'  0",  t  =  llh  25'*  4s;  required  A  and  £. 

We  find  A  =  241°  53'  33".2,  £  =  126°  25'  6".6;  the  star  is 
helow  the  horizon,  and  its  negative  altitude,  or  depression,  is 
h  =  —  36°  25'  6".6. 

If  the  zenith  distance  of  the  same  star  is  to  he  frequently  com- 
puted on  the  same  night  at  a  given  place,  it  will  be  most  readily 
done  by  the  following  method.  In  the  first  equation  of  (14) 

substitute 

cos  t  =  I  —  2  sin2  }  t 
then  we  have 

cos  C  =  cos  (y*r8)  —  2  cos  <f>  cos  d  sin2  £  t 

where  <p  ^  d  signifies  either  <p  —  d  or  d  —  ^>,  and  if  d  >  <p  the  latter 
form  is  to  be  used.  Subtracting  both  members  from  unity,  we 
obtain 

sin2  £  C  =  sin2  \  (<p  *r  d)  -f-  cos  <p  cos  d  sin2  ?  t 
Now  let 

m  =  i/cos  <p  cos  d 
n  =  sin  £  (p  *r  d) 
then  we  have 


and  hence,  by  taking  an  auxiliary  N  such  that 

TVT      m   - 
tan  J\  =  —  sin  £  t 

n 
we  have 

m 


sn        = 


cos  iv      sin  JV 

The  second  form  for  sin  J  f  will  be  more  precise  than  the  first 
when  sin  N  is  greater  than  cos  N. 

The  quantities  m  and  n  will  be  constant  so  long  as  the  decli- 
nation does  not  vary. 

15.  If  the  parallactic  angle  q  (Art.  11)  and  the  zenith  distance 

VOL.  I.— :i 


34 


THE    CELESTIAL    SPHERE. 


(19) 


£  are  required  from  the  data  p,  <5,  and  t,  they  may  be  found 
from  the  equations 

cos  C  —  sin  <p  sin  5  -j-  cos  <p  cos  5  cos  t          \ 
sin  C  cos  q  =  sin  50  cos  5  —  cos  <p  sin  d  cos  f 
sin  C  sin  q  =  cos  ^  sin  t 

which  are  adapted  for  logarithms  as  follows : 

n  sin  JV=  cos  <p  cos  t 

n  cos  JV=  sin  <p 

cos  C  =  n  sin  (5  -f-  -ZV) 
sin  C  cos  <?  =  n  cos  (5  -f  JV) 
sin  C  sin  </  =  cos  <p  sin  t 

or,  eliminating  n,  thus : 

tan  N=  cot  ^  cos  # 

,    .       •     tan  t  sin  N 

tan  C  sin  q  =  — 

sin  (5  -f  JV) 

tan  C  cos  <?  —  cot  (<J  -f-  JV) 

When  this  last  form  is  employed  in  the  case  of  a  star  which 
has  been  observed  above  the  horizon,  tan  f  is  known  to  be  posi- 
tive, and  there  is  no  ambiguity  in  the  determination  of  q.  This 
form  is,  therefore,  the  most  convenient  in  practice. 

If  ^  A,  and  q  are  all  required  from  the  data  d,  t,  and  p,  we 
have,  by  Gauss's  equations-, 


sin  J  C  sin  J  (A  -f-  0)  —  sin  |  £  cos  J  O  -(- 
sin  J  C  cos  i  (A  -f  #)  =  cos  $  £  sin  J  (<p  — 
cos  J  C  sin  £  (A  •—  #)  =  sin  J  £  sin  J  (^  -f- 
cos  A  C  cos  J  (A  —  0)  =  cos  *  f  cos  J  p  — 


(21) 


Fig.  4. 


16.  When  the  altitude,  azimuth,  and  parallactic  angle  of  known 
stars  are  to  be  frequently  computed  at  the 
same  place,  the  labor  of  computation  is 
much  diminished  by  an  auxiliary  table  pre- 
pared for  the  latitude  of  the  place  accord- 
ing to  formulae  proposed  by  Gauss.  A 
specimen  of  such  a  table  computed  for  the 
latitude  of  the  Altona  Observatory  will  be 
found  in  "Schumachers  Hiilfstofdn,  neu 
herausg.  v.  Warnstorif."  The  requisite 
formulae  are  readily  deduced  as  follows : 

Let  the  declination  circle  through  the  object   0,  Fig.  4,  be 
produced  to  intersect  the  horizon  in  F.     By  the  diurnal  motion 


SPHERICAL    CO-ORDINATES.  35 

the  point  F  changes  its  position  on  the  horizon  with  the  time ; 
but  its  position  depends  only  on  the  time  or  the  hour  angle 
ZPO,  and  not  upon  the  declination  of  0.  The  elements  of  the 
position  of  F  may  therefore  be  previously  computed  for  succes- 
sive values  of  t. 

We  have  in  the  triangle  PFS,  right-angled  at  S,  FPS=t, 
PS  =  180°  —  <p\  and  if  we  put 

a  =  -F£,        B  =  PF—9Q°,       r  =  ISO0  — PFS. 
we  find 

tan  a  =  sin  <f>  tan  t,     tan  B  =  cot  <f>  cos  t,     cot  f  =.  sin  B  tan  t 
We  have  now  in  the  triangle  HOF,  right-angled  at  H, 
B  +  d=OF,  r  =  HFO,  h  =  OH, 

and  if  we  put 

u  =  HF  =  HS—FS  =  A—&, 

we  find 

tan  u  =  cos  f  tan  (B  -f  <5)  J.  =  H  -j-  w 

sin  A  =  sin  Y  sin  (jB  -}-  #)          or,  tan  A  =  tan  Y  sin.w. 

To  find  the  parallactic  angle  q=POZ,  we  have  in  the  triangle 

HOF 

tan  q  =  cot  Y  sec  (5  -f-  fl) 

In  the  Gaussian  table  for  Altona  as  given  in  the  "Hiilfstafeln" 
we  find  five  columns,  which  give  for  the  argument  t,  the  quan- 
tities a,  -#,  log  cos  Y,  log  sin  ft  l°g  co^  r>  ^ne  ^ast  three  under 
the  names  log  (7,  log  D,  and  log  _Z£,  respectively.  With  the  aid 
of  this  table,  then,  the  labor  of  finding  any  one  of  the  quan- 
tities A,  A,  q  is  reduced  to  the  addition  of  two  logarithms, 
namely: 

tan  u  =  C  tan  (B  +  d)  s\nh  =  D  sin  (B  -f  S} 

A  =  &  +  u  tan  q  =  E  sec  (B  -f  8) 

The  formulae  for  the  inverse  problem  (of  Art.  10)  may  also  be 
found  thus.  Let  G  be  the  intersection  of  the  equator  and  the 
vertical  circle  through  0,  and  put  B  =  HG,  u  =DG,  &=  QG, 
Y=  ZGQ;  then  we  readily  find 

tan  a  =  sin  <p  tan  A,     tan  B  =  cot  <f>  cos  J.,     cot  y  =  8\n  B  tan  ^L 

which  are  of  the  same  form  as  those  given  above,  with  the  ex- 
change of  A  for  t.  Hence  the  same  table  gives  also  the  elements 
of  the  point  G,  by  entering  with  the  argument  "azimuth,"  ex- 
pressed in  time,  instead  of  the  hour  angle.  We  then  have  t  = 


36  THE    CELESTIAL   SPHERE. 


),  and  if  we  here  put  u=DGr  =  &  —  t,  we  have  from  the 
triangle  GDO 

sin  d  =  sin  f  sin  (h  —  B)  tan  u  =  cos  Y  tan  (A  — B) 

or,  employing  the  notation  of  the  table, 

tan  u  =  C  tan  (h  —  B)  sin  <?  =  D  sin  (h  —  B) 

17.  To  find  the  zenith  distance  and  azimuth  of  a  star,  when  on  the 
six  hour  circle. — Since  in  this  case  t  =  6A  —  90°,  the  triangle  PZO, 
Fig.  4,  is  right-angled  at  P,  and  gives  immediately 

cos  ZO  =  cos  PZ  cos  PO  cot  PZO  =  sin  PZ  cot  PO 

or,  since  PZO  =  180°  —A,  and  cot  PZO=  -  cot  A, 

cos  C  =  sin  <p  sin  5  cot  <A  =  —  cos  <p  tan  # 

But  if  the  star  is  on  the  six  hour  circle  east  of  the  meridian, 
we  must  put  t  =  18A=270°  and  PZO=A  — 180°  ;  hence  for  this 

case 

cot  A  =  -\-  cos  <p  tan  d 

A  more  general  solution,  however,  is  obtained  from  the  equa- 
tions (14),  by  putting  cos  /  =  0,  sin  t  =  db  1,  whence 

cos  C  =       sin  <f  sin  d  ^ 

sin  C  cos  A  =  —  cos  <p  sin  3  >     (22) 

sin  C  sin  A  =  ±  cos  5  J 

the  lower  sign  in  the  last  equation  being  used  when  the  star  is 
east  of  the  ineridian. 

EXAMPLE. — Required  the  zenith  distance  and  azimuth  of  Sirius, 
§  =  -  16°  31'  20",  when  on  the  six  hour  circle  eastpf  the  meri- 
dian at  the  Cape  of  Good  Hope,  <p  =  -  33°  56'  3".  We  find 

log  (—  cos  d)  =  log  sin  C  sin  A  =-^9.9816870 
log  (—  cos  <f>  sin  S)  =  log  sin  C  cos  A  =   9.3728204 

A  =   283°49'34".9 
log  sin  A  =  9.9872302 
log  sin  C  =   9.9944568 
log  sin  <p  sin  3  =  log  cos  C  =   9.2007309 

C  =    80°  51' 55" 


SPHERICAL    CO-ORDINATES. 


37 


Fig.  5. 


18.  To  find  the  hour  angle,  azimuth,  and  zenith  distance  of  a  given 
star  at  its  greatest  elongation. — In  this  case  the  vertical  circle 
ZS,  Fig.  5,  is  tangent  to  the  diurnal  circle, 
SA,  of  the  star,  and  is,  therefore,  perpendicular 
to  the  declination  circle  PS.  The  right  triangle 
PZS  gives,  therefore, 

tan  <p 


cos  t  = 


sm  A  = 


cos  C  — 


tan  d 
cos  d 
cos  <p 
sin  <p 
sin  5 


(23) 


If  d  and  <p  are  nearly  equal,  each  of  the  quantities  cos  t,  sin  A, 
and  cos  £  will  be  nearly  equal  to  unity,  and  a  more  accurate 
solution  for  that  case  will  then  he  as  follows  : 

Subtract  the  square  of  each  from  unity  ;  then  we  have 

-j-  y>}  sin  (d  —  <p) 


sin2  t  = 


cos2  J.  = 


sn2  C  = 


tan2  d 
cos2  ^  —  cos2 

cos2  <f> 
sin2  5  —  sin2 


sn 


sn 


-f- 


sin2<5 
sin    <J 


COS2  (f> 

sin  (ft  4-  ^    sin 


sin2  d 


sn2 


Hence  if  we  put 

k  = 
we  shall  have 

* 


sin  (d  — 


sin  t  = 


cos  <p  sin  d 


cos  A 


cos 


sin  C  = 


sin  d 


(24) 


19.  To  find  the  hour  angle,  zenith  distance,  and  parallactic  angle  of 
a  given  star  on  the  prime  vertical  of  a  given  place. 

In  this  case,  the  point  0  in  Fig.  1  being  in  the  circle  WZE, 
the  angle  PZO  is  90°,  and  the  right  triangle  PZO  gives 

tan  d 


cos  t  = 


tan 


sin  d 

cos  C  =  — — 
sin  <p 

cos  <f> 
^         cos  d 


(25) 


38  THE    CELESTIAL    SPHERE. 

If  d  is  but  little  less  than  y>,  the  star  will  be  near  the  zenith, 
and,  as  in  the  preceding  article,  we  shall  obtain  a  more  accurate 
solution  as  follows : 

Put 

k  =  |/[sin  (<p  -j-  3)  sin  (y  —  <5)] 
then 

k  k  k 

sin  t  =  -r T         sin  C  ==  — —  cosq  — (26) 

sin  (p  cos  3  sin  y  cos  d 

We  may  also  deduce  the  following  convenient  and  accurate 
formulae  for  the  case  where  the  star's  declination  is  nearly  equal 
to  the  latitude  [see  Sph.  Trig.  Arts.  60,  61,  62]  : 


tan 


=,      </ta 

\  Ua 


tan  \  (c£>  -j- 
tan  (45°  —  \  q)  =  j/[tan  J  (9?  -f-  5)  tan.£  (<p  —  «5)] 


(27) 


If  d  >  ^>,  these  values  become  imaginary;  that  is,  the  star  can- 
not cross  the  prime  vertical. 

EXAMPLE. — Required  the  hour  angle  and  zenith  distance  of  the 
star  12  Caniun  Venaticorum  (d  =  -j-  39°  5'  20")  when  on  the  prime 
vertical  of  Cincinnati  (f  =  =  +  39°  5'  54"). 

^  _  ^  =      o°  0'  34"  %  (<?  —  S)=      0°  0'  17" 

p  -f  fl  =    78  11  14  J  (^  —  5)  =    39  5  37 

log  sin  (p  —  5)  6.21705  log  tan  \  ($-—  d)  5.91602 

log  sin  (<p  -f  3)  9.99070  log  tan  i  (>  -f  5)  9.90982 

2)6.22635  2)6.00620 

log  tan  $  f          8.11318  log  tan  J  C  8.00310 

}*  =  0°  44'  36".6  }  C  =  0°  34'  37".3 

t  =  1°  29'  13".2  C  =  1°    9'  14".6 

=  0*  5-  56'.88 

20.    To  find  the  amplitude  and  hour  angle  of  a  given  star  when  in 
the  horizon. — If  the  star  is  at  //,  Fig.  1,  we  have  in  the  triangle 
PHN,    right-angled   at  N,  PN=  ^  HPN  =  =  180°  —  t,  PH  = 
90°    -  3 ;   and  if  the  amplitude  WH  is  denoted  by  a,  we  have 
HN=  90°  —  a.     This  triangle  gives,  therefore, 

sin  a  =  sec  <f>  sin  3  1 

cos  t  =  —  tan  <f>  tan  3  ) 


SPHERICAL    CO-ORDINATES.  39 

21.  Given  the  hour  angle  (t)  of  a  star,  to  find  its  right  ascension  (a). 
— Transformation  from  our  second  system  of  co-ordinates  to  the 
third. 

There  must  evidently  be  given  also  the  position  of  the  meridian 
with  reference  to  the  origin  of  right  ascensions.  Suppose  then 
in  Fig.  1  we  know  the  right  ascension  of  the  meridian,  or  VQ 
=  0,  then  we  have  VD  ==  VQ  —DQ,  that  is, 

a  =  0  —  t 

Conversely,  if  a  and  0  are  known,  we  have 

t  =  0  —  a 

The  methods  of  finding  0  at  a  given  time  will  be  considered 
hereafter. 

22.  Given  the  zenith  distance  of  a  known  star  at  a  given  place,  to 
find  the  stars  hour  angle,  azimuth,  and parallactic  angle. 

In  this  case  there  are  given  in  the  triangle  POZ,  Fig.  1,  the 
three  sides  ZO  =  f ,  PO  =  90°  —  d,  PZ  =  90°  —  <p,  to  find 
the  angles  ZPO  =  t,  PZO  =  180°  —  A,  and  POZ=q.  The 
formula  for  computing  an  angle  B  of  a  spherical  triangle  ABC, 
whose  sides  are  a,  b,  c,  is  either 

sin  1  B  =  J  /  "i"  (« -a)8in  (»  -  e)  \ 
\   V  sin  a  sin  c  I 

cos  \  B  =  J  /8in*sin(s-6)\ 
\    \        sin  a  sin  c        / 

or  tn.n  i  K  —     /  /  Bin  (*  -  a)  sin  (s  -  c)  \ 

\   \        sin  s  sin  (5  —  b)         I 

in  which  s  =  \  (a  +  b  -f  c).  We  have  then  only  to  suppose  B 
to  represent  one  of  the  angles  of  our  astronomical  triangle,  and 
to  substitute  the  above  corresponding  values  of  the  sides,  to  ob- 
tain the  required  solution. 

This  substitution  will  be  carried  out  hereafter  in  those  cases 
where  the  problem  is  practically  applied. 

23.  Given  the  declination  (d)  and  the  right  ascension  (a)  of  a  star, 
and  the  obliquity  of  the  ecliptic  (e),  to  find  the  latitude  (/?)  and  the  longi- 
tude (X]  of  the  star. — Transformation  from  the  third  system  of  co- 
ordinates to  the  fourth. 

The  solution  of  this  problem  is  similar  to  that  of  Art.  10. 


40 


THE    CELESTIAL    SPHERE. 


The  analogy  of  the  two  will  be  more  apparent  if  we  here  repre- 
sent the  sphere  projected  on  the  plane  of 
the  equator  as  in  Fig.  6,  where  VB  UC  is 
the  equator,  P  its  pole ;  VA  U  the  ecliptic, 
P'  its  pole,  and  consequently  CP'PB  the 
solstitial  colure;  POD,  P'OL,  circles  of 
u  declination  and  latitude  drawn  through  the 
star  0.  Since  the  angle  which  two  great 
circles  make  with  each  other  is  equal  to 
the  angular  distance  of  their  poles,  we  have 
PPr  =  e ;  and  since  the  angle  P'PO  is 

measured   by   CD  and  PP'O  by  AL,  w^e  have  in  the  triangle 

PP'O 


P'PO, 

90°  +  a, 


PP'O, 

90°  —  A, 


P'O, 

90°  _ 


PO, 

90°  — 


PP' 


which,  substituted  respectively  for 

A,  B         .         a,  6, 

in  the  general  equations  (31),  Art.  10,  give 

sin  /5  =  cos  e  sin  d  —  sin  e  cos  d  sin  a 
cos  /5  sin  A  =  sin  e  sin  d  -j-  cos  e  cos  <5  sin  a 
cos  £  cos  A  =  cos  d  cos  a 


(29) 


which  are  the  required  formulae  of  transformation.     Adapting 
for  logarithmic  computation,  we  have 


m  sin  M  =  sin  d 

m  cos  M  =  cos  d  sin  a 

sin  /3  —  m  sin  (Jf —  e) 
cos  ft  sin  A  —  m  cos  (M  —  e) 
cos  0  cos  A  —  cos  d  cos  a 


in  which  m  is  a  positive  number. 

A  still  more  convenient  form  is  obtained  by  substituting 


(30) 


m 


cos 


cos 
m 


by  which  we  find 


SPHERICAL    CO-ORDINATES.  41 


A*  sin  Jf  =  tan  d 
k  cos  Jf  =  sin  a 
A' sin    A  =  cos  (Jf —  e) 
A'  cos  A  —  cos  Jf  cot  a 
tan  y3  —  sin  A  tan  (Jf  —  e) 

cos  y9  sin  A       cos  (Jf  —  e) 


(31) 

um  yj  -=  sin  A  ian  ^o.  —  e) 

cos  /5  sin  A        fns  /"/Tf  —  ?"\ 
For  verification:     CQS  ^  sin  g 


EXAMPLE. — Given  o,  a,  and  e  as  below,  to  find  /9  and  L  Com- 
putation by  (31). 

9  =  —  16°  22'  35".45  log  sin  A             ^8.0897286 

a=         6    3329.30  log  tan  (Jf—e)  1.4114658 

e  =      23    27   31  .72  log  tan  /?            n9.5011944 

log  tan  d  =  log  A-  sin  M  n9.4681562  ,9  =  —  17°  35'  37".51 

log  sin  a  =  log  A-  cos  M     9.0577093 

M=  —  68°  45'  41".87  Verification. 

M  —  e  =  —  92    13   13  .59  log  cos  ,?  sin  /I  n8.0689234 

log  cos  (S  sin  a     9.0397224 

log  cos  Jf     9.5590070  cos(Jf-e) 

i  r\  nonion/1  o  ~       ilaAIAoAvLv. 

log  cot  a       0.9394396  cos  M 

log  A'  cos  X    0.4984463 
logcos(Jf—  e)  =  log  k'  sin/l  n8.5882080 
A  =  359°  17'  43".91 

Tables  for  facilitating  the  above  transformation,  based  upon 
the  same  method  as  that  employed  in  Art.  16,  are  given  in  the 
American  Ephemeris*  and  Berlin  Jahrbuch.  The  formulae  there 
used  may  be  obtained  from  Fig.  6,  in  which  the  points  J^ancl  G 
are  used  precisely  as  in  Fig.  4  of  Art.  16. 

24.  If  we  denote  the  angle  at  the  star,  or  P'OP,  by  90°  —  E, 
the  solution  of  the  preceding  problem  by  Gauss's  Equations  is 


cos (45°  —  J0)sinJ  (•#+*)  =  SHI  [45°  —  £(£_ a)]  sin  (45° -f  J  a) 
cos  (45°  —  i  /3)  cos  $(E  -f  A)  =  cos  [45°  —  I  (t  +  S j]  cos  (45°  -f  i  a) 
sin  (45°  —  i/5)sin  K^J— A)  =  sin  [45°  —  £  (e  +  *)] cos (45°  -f  *«) 
sin  (45°  — ^/3)  cos  K^— A)  =  cos  [45°  —  J(e  —  (5)]  sin  (45°  + i  a) 


(32) 


25.  If  the  angle  at  the  star  is  required  when  the  Gaussian 
Equations  have  not  been  employed,  we  have  from  the  triangle 
POP',  Fig.  6,  putting  P'OP  =  y  =  90°  —  E, 

*  In  the  volumes  1855-64  only. 


42 


THE    CELESTIAL    SPHERE. 


cos  ft  cos  y  =  cos  e  cos  d  -j-  sin  e  sin  d  sin  a 
cos  ft  sin  iy  =  sin  e  cos  a 


or,  adapted  for  logarithms, 


n  sin  N  —  sin  e  sin  a 

n  cos  -ZV  =  cos  e 
cos  ft  cos  TJ  =  n  cos  (.2V  — 
cos  /3  sin  ^  =  sin  e  cos  a 


(33) 


26.  6?wen  /Ae  latitude  (/?)  «wd  longitude  (X)  of  a  star,  ami  the 
obliquity  of  the  ecliptic  (s),  to  find  the  declination  and  right  ascension 
of  the  star. 

By  the  process  already  employed,  we  derive  from  the  triangle 
PP'O,  Fig.  6,  for  this  case, 


sin  d  =  cose  sin  ft  -f-  sin  e  cos  ft  sin  A 
cos  d  sin  a  =  —  sin  e  sin  ft  -j-  cos  e  cos  ,3  sin  A 
cos  d  cos  a  =  cos  ft  cos  A 


(34) 


which,  it  will  he  observed,  may  be  obtained  from  (29)  by  inter- 
changing a  with  A,  and  d  with  /5,  and  at  the  same  time  changing 
the  sign  of  e,  that  is,  putting  —  £  for  e,  and,  consequently,  —  sin  e 
for  sin  e. 

For  logarithmic  computation,  we  have 


m  sin  M  =  sin  /3 

m  cos  Jf  =  cos  /3  sin  A 

sin  <5  z=  7?i  sin  (Jlf  -j-  e) 
cos  5  sin  a  =  m  cos  (  Jf  -j-  e) 
cos  <5  cos  a  =  cos  /3  cos  A 


or  the  following,  analogous  to  (31)  : 


A1  sin  M  =  tan  ft 

k  cos  M  =  sin  A 

A''  sin    a  =  cos  (^f  -j-  e) 

A-'  cos  a  =  cos  Jf  cot  /I 

tan  d  =  sin  a  tun  (M  -j-  e) 


cos  5  sin  a       cos  (Jf  -f  e) 

Jrbf  verification  :   -      0   .  —  r—  ~~        —  1^~ 
cos  /3  sin  A  cos  Jf 


(35) 


(36) 


27.  The  angle  at  the  star,  POP',  being  denoted,  as  in  Art.  24, 


RECTANGULAR    CO-ORDINATES.  43 

by  90°  —  E,    the    solution    of  this  problem   by   the    Gaussian 
Equations  is 


sin  (45°—  J  8)  sin  }  (E  -f  a)  =  sin  [45°—  £  (e  +  /?)]  sin  (45°+  J  /I) 
sin  (45°—  J  <5)  cos  J  (J£-f  «)  =  cos  [45°—  1  (*—£)]  cos(45°-f  i  A) 
cos  (45°—  J  <5)  sin  J  (^  —  a)  ==  sin  [45°—  i  (e  —  /?)]  cos  (45°+  *  A) 
cos  (45°—  \  <5)  cos  J  (^  —  a)  =  cos  [45°—  i  (e  -f  /9)]  sin  (45°-j-  \  X) 


(37) 


28.  But   if  the   angle   y  =  90°     -  E  is    required   when   the 
Gaussian  Equations  have  not  been  employed,  we  have  directly 

cos  d  cos  TJ  —  cos  e  cos  ft  —  sin  e  sin  /9  sin  A 
cos  d  sin     =  sin  s  cos  A 


or,  adapted  for  logarithms, 

N=  sin  e  sin  X 

N=  cos  £ 

(38) 


n  sn      =  sn  e  sn 

n  cos  JY=  cos  £ 
cos  d  cos  7]  =  n  cos  (iV  -f  /3) 
cos  ^  sin  T  =  sin  £  cos  A 


29.  J^or  ^Ae  sun,  we  may,  except  when  extreme  precision  is 
desired,  put  /?  =  0,  and  the  preceding  formulae  then  assume  very 
simple  forms.     Thus,  if  in  (34)  we  put  sin  /3  =  0,  cos  ft  =  1,  we 
find 

sin  d  =  sin  £  sin  A 
cos  d  sin  a  =  cos  £  sin  A 
cos  3  cos  a  =  cos  A 

whence  if  any  two  of  the  four  quantities  <?,  a,  ^,  e  are  given,  we 
can  deduce  the  other  two. 

RECTANGULAR    CO-ORDINATES. 

30.  By  means  of  spherical  co-ordinates  we  have  expressed 
only  a  star's  direction.     To   define   its   position   in   space   com- 
pletely, another  element  is  necessary,  namely,  its  distance.     In 
Spherical  Astronomy  we  consider  this  element  of  distance  only 
so   far   as   may  be    necessary   in   determining   the    changes  of 
apparent  direction  of  a  star  resulting  from  a  change  in  the  point 
from  which  it  is  viewed.     For  this  purpose  the  rectangular  co- 
ordinates of  analytical  geometry  may  be  employed. 

Three  planes  of  reference  are  taken  at  right  angles  to  each 
other,  their  common  intersection,  or  origin,  being  the  point  of 


44  THE    CELESTIAL    SPHERE. 

observation ;  and  the  star's  distances  from  these  planes  are 
denoted  by  #,  ?/,  and  z  respectively.  These  co-ordinates  are 
respectively  parallel  to  the  three  axes  (or  mutual  intersections 
of  the  planes,  taken  two  and  two),  and  hence  these  axes  are 
called,  respectively,  the  axis  of  z,  the  axis  of  y,  and  the  axis  of  z. 
The  planes  are  distinguished  by  the  axes  they  contain,  as  "the 
plane  of  xy"  the  "plane  of  xz"  the  "plane  of  yz."  The  co- 
ordinates may  be  conceived  to  be  measured  on  the  axes  to 
which  they  belong,  from  the  origin,  in  two  opposite  directions, 
distinguished  by  the  algebraic  signs  of  plus  and  minus,  so  that 
the  numerical  values  of  the  co-ordinates  of  a  star,  together  with 
their  algebraic  signs,  fully  determine  the  position  of  the  star  in 
space  without  ambiguity.  • 

Of  the  eight  solid  angles  formed  by  the  planes  of  reference, 
that  in  which  a  star  is  placed  will  always  be  known  by  the  signs 
of  the  three  co-ordinates,  and  in  one  only  of  these  angles  will 
the  three  signs  all  be  plus.  This  angle  is  usually  called  the  frst 
angle.  To  simplify  the  investigations  of  a  problem,  we  may,  if 
we  choose,  assume  all  the  points  considered  to  lie  in  the  first 
angle,  and  then  treat  the  equations  obtained  for  this  simplest 
case  as  entirely  general ;  for,  by  the  principles  of  analytical 
geometry,  negative  values  of  the  co-ordinates  which  satisfy  such 
equations  also  satisfy  a  geometrical  construction  in  which  these 
co-ordinates  are  drawn  in  the  negative  direction. 

The  polar  co-ordinates  of  analytical  geometry  (of  three  dimen- 
sions) when  applied  to  astronomy  are  nothing  more  than  the 
spherical  co-ordinates  we  have  already  treated  of,  combined 
with  the  element  distance ;  and  the  formulae  of  transformation 
from  rectangular  to  polar  co-ordinates  are  nothing  more  than 
the  values  of  the  rectangular  co-ordinates  in  terms  of  the  dis- 
tance and  the  spherical  co-ordinates.  For  the  convenience  of 
reference,  we  shall  here  recapitulate  these  formulae,  with  special 
reference  to  our  several  systems  of  spherical  co-ordinates. 

31.  We  shall  find  it  useful  to  premise  the  following 
LEMMA. —  The  distance  of  a  point  in  space  from  the  plane  of  any 
great  circle  of  the  celestial  sphere  is  equal  to  its  distance  from  the  centre, 
of  the  sphere  multiplied  by  the  cosine  of  its  angular  distance  from  the 
pole  of  that  circle;  and  its  distance  from  the  axis  of  the  circle  is  eqml  to 
Us  distance  from  the  centre  of  the  sphere  multiplied  by  the  sine  of  its 
angular  distance  from  the  pole. 


RECTANGULAR    CO-ORDINATES.  45 

For,  let  AB,  Fig.  7,  be  the  given  great  circle  orthographi- 
cally  projected  upon  a  plane  passing  through  its  axis  OP  and 
the  given  point  C;  P  its  pole.  The  dis- 
tance of  the  point  C  from  the  plane  of  the 
great  circle  is  the  perpendicular  CD;  CE 
is  its  distance  from  the  axis ;  CO  its  dis- 
tance from  the  centre  of  the  sphere ;  and 
the  angle  COP  the  angular  distance  from 
the  pole.  The  truth  of  the  Lemma  is, 
therefore,  obvious  from  the  figure. 

32.  The  values  of  the  rectangular  co-ordinates  in  our  several 
systems  may  be  found  as  follows  : 

First  system.. — Altitude  and  azimuth. — Let  the  primitive  plane, 
or  that  of  the  horizon,  be  the  plane  of  xy ;  that  of  the  meridian, 
the  plane  of  xz ;  that  of  the  prime  vertical,  the  plane  of  yz. 
The  meridian  line  is  then  the  axis  of  x;  the  east  and  west  line, 
the  axis  of  y ;  and  the  vertical  line,  the  axis  of  z.  Positive  x 
will  be  reckoned  from  the  origin  towards  the  south,  positive  y 
towards  the  west,  and  positive  z  towards  the  zenith.  The  first 
angle,  or  angle  of  positive  values,  is  therefore  the  southwest 
quarter  of  the  hemisphere  above  the  plane  of  the  horizon.  Let 
Z,  Fig.  8,  be  the  zenith,  S  the  south  point,  W  the  ris.  8. 

west  point  of  the   horizon.     These  points  are       , 
respectively  the  poles  of  the  three  great  circles 
of  reference ;    if,  then,  A  is  the  position  of  a 
star  on  the  surface  of  the  sphere  as  seen  from 
the  centre  of  the  earth,  and  if  we  put 

h  =  altitude  of  the  star  =  AH, 
A  =  azimuth  p  SH, 

A  =r  its  distance  from  the  centre  of  the  sphere 

we  have  immediately,  by  the  preceding  Lemma, 

x  =  A  cos  AS,          y  =  A  cos  A  W,          z  =  A  cos  AZ, 
which,  by  considering  the  right  triangles  AHS,  AHW,  become 

x  —  A  cos  h  cos  A  ^ 

y  =-:J  cos  h  sin  A  V      (39) 

z  =  A  sin  h  ) 

These  equations  determine  the  rectangular  co-ordinates  x,y,z, 


46  THE    CELESTIAL    SPHERE. 

when  the  polar  co-ordinates  J,  h,  A  are  given.  Conversely,  if 
x,  #,  and  z  are  given,  we  may  find  J,  A,  and  A  ;  for  the  first  two 
equations  give 


if 

tan  A  =  - 
x 

and  then  we  have 

A  sin  h  =  z 

A  cos  h  =      X      =~      y 

cos  A         sin  A 

whence  A  and  A.     Or,  by  adding  the  squares  of  the  first  two 
equations,  we  have 


A  cos  h  —  Vx1  -j- 
whence 

z 


tan  A  = 


and  the  sum  of  the  squares  of  the  three  equations  gives 


Second  system..  —  Declination  and  hour  angle.  —  Let  the  plane  of 
the  equator  be  the  plane  of  xy  ;  that  of  the  meridian,  the  plane 
of  xz;  that  of  the  six  hour  circle,  the  plane  of  yz.  In  the  pre- 
ceding figure,  let  Z  now  denote  the  north  pole,  S  that  point  of 
the  equator  which  is  on  the  meridian  above  the  horizon  and 
from  which  hour  angles  are  reckoned,  TV  the  west  point.  Posi- 
tive x  will  be  reckoned  towards  $,  positive  y  towards  the  west, 
positive  z  towards  the  north.  If  then  A  is  the  place  of  a  star  on 
the  sphere  as  seen  from  the  centre,  and  we  put 

d  =  the  star's  declination  =  AH, 
t  =  "  hour  angle  =  SH, 
A  =  "  distance  from  the  centre, 

and  denote  the  rectangular  co-ordinates  in  this  case  by  x1  ',  y',  zf, 

we  have 

x'  —  A  cos  d  cos  t  "| 

y'  =  A  cos  d  sin  t  \      (40) 

z'  =  A  sin  (5  J 

Third  system.  —  Declination  and  right  ascension.  —  Let  the  plane 
of  the  equator  be  the  plane  of  xy  ;  that  of  the  equinoctial  colure, 
the  plane  of  xz;  that  of  the  solstitial  colure,  the  plane  of  yz. 


RECTANGULAR    CO-ORDINATES.  47 

The  axis  of  x  is  the  intersection  of  the  planes  of  the  equator 
and  equinoctial  colure,  positive  towards  the  vernal  equinox ;  the 
axis  of  y  is  the  intersection  of  the  planes  of  the  equator  and  sol- 
stitial colure,  positive  towards  that  point  whose  right  ascension 
is  -j-90° ;  and  the  axis  of  z  is  the  axis  of  the  equator,  positive 
towards  the  north.  If  then,  in  Fig.  8,  Z  is  the  north  pole,  W 
the  vernal  equinox,  A  a  star  in  the  first  angle,  projected  upon 
the  celestial  sphere,  and  we  put 

d  =  declination  of  the  star  =  AH, 
a  =  right  ascension  "  =  WH, 
A  =  distance  from  the  centre, 

while  x",  y" ,  z"  denote  the  rectangular  co-ordinates,  we  have 
x"  =  A  cos  A  W,         y"  =  A  cos  AS,        z"  =  A  cos  AZ, 

which  become 

xff  =  A  cos  8  cos  a  "j 

y"  =  A  cos  8  sin  a  V      (41) 

z"  =  A  sin  8  ) 

Fourth  system. — Celestial  latitude  and  longitude. — Let  the  plane 
of  the  ecliptic  be  the  plane  of  xy ;  the  plane  of  the  circle  of 
latitude  passing  through  the  equinoctial  points,  the  plane  of  xz  ; 
the  plane  of  the  circle  of  latitude  passing  through  the  solstitial 
points,  the  plane  of  yz.  The  positive  axis  of  x  is  here  also  the 
straight  line  from  the  centre  towards  the  vernal  equinox ;  the 
positive  axis  of  y  is  the  straight  line  from  the  centre  towards  the 
north  solstitial  point,  or  that  whose  longitude  is  +90° ;  and  the 
positive  axis  of  z  is  the  straight  line  from  the  centre  towards 
the  north  pole  of  the  ecliptic. 

If  then,  in  Fig.  8,  Z  now  denotes  the  north  pole  of  the  ecliptic, 
W  the  vernal  equinox,  A  the  star's  place  on  the  sphere,  and 
we  put 

ft  —  latitude  of  the  star     =  AH, 

A  =  longitude  of  the  star  =  WH, 

A  =  distance  of  the  star  from  the  centre, 

and  x'",  ynt ,  z'n ,  denote  the  rectangular  co-ordinates  for  this 
system,  we  have 

x"'  =  A  cos  /5  cos  A  ^ 

y'"  =  A  cos  /9  sin  A  I      (42) 

z'"  =  A  sin 


48  THE    CELESTIAL    SPHERE. 


TRANSFORMATION   OF   RECTANGULAR   CO-ORDINATES. 

33.  For  the  purposes  of  Spherical  Astronomy,  only  the  most 
simple  cases  of  the  general  transformations  treated  of  in  analy- 
tical geometry  are  necessary.  We  mostly  consider  hut  two  cases : 

First.  Transformation  of  rectangular  .co-ordinates  to  a  new  origin, 
without  changing  the  system  of  spherical  co-ordinates. 

The  general  planes  of  reference  which  have  been  used  in  this 
chapter  may  he  supposed  to  he  drawn  through  any  point  in  space 
without  changing  their  directions,  and  therefore  without  changing 
the  great  circles  of  the  infinite  celestial  sphere  which  repre- 
sent them.  We  thus  repeat  the  same  system  of  spherical  co-ordi- 
nates with  various  origins  and  different  systems  of  rectangular 
co-ordinates,  the  planes  of  reference,  however,  remaining  always 
parallel  to  the  planes  of  the  primitive  system. 

The  transformation  from  one  system  of  rectangular  co-ordi- 
nates to  a  parallel  system  is  evidently  effected  hy  the  formulae 

•*'i  =  x»  -f  a  \ 

Vi  =  y*  +  b  (43) 


in  which  xv  y^  zl  are  the  co-ordinates  of  a  point  in  the  primitive 
system ;  z2,  y2,  Z2  the  co-ordinates  of  the  same  point  in  the  new 
system ;  and  a,  6,  c  are  the  co-ordinates  of  the  new  origin  taken 
in  the  first  system. 

As  we  have  shown  how  to  express  the  values  of  x^  yv  zl  and 
of  #2,  3/2,  Z2  in  terms  of  the  spherical  co-ordinates,  we  have  only 
to  substitute  these  values  in  the  preceding  formulae  to  obtain  the 
general  relations  between  the  spherical  co-ordinates  correspond- 
ing to  the  two  origins.  This  is,  indeed,  the  most  general  method 
of  determining  the  effect  of  parallax,  as  will  appear  hereafter. 

Fig>  9t  Second.    Transformation    of  rectangular    co- 

Zj  z  ordinaies  when  the  system  of  spherical  co-ordi- 

J  nates  is  changed  but  the   origin  is  unchanged. 

&  /b^ A        This  amounts  to  changing  the  directions  of 
the  axes.     The  cases  which  occur  in  practice 
~*1  are  chiefly  those  in  which  the  two  systems 
*  have  one  plane  in  common.     Suppose  this 
plane  is  that  of  xz,  and  let  OX,  OZ,  Fig.  9,  be 
the  axes  of  x  and  z  in  the  first  system; 


RECTANGULAR    CO-ORDINATES.  49 

OZlt  the  axes  of  xl  and  zl  in  the  new  system.  Let  A  be  the 
projection  of  a  point  in  space  upon  the  common  plane ;  and 
let  x  =  AB,  z  =  OB,  xl=ABv  z,  =OBr  The  distance  of  the 
point  from  the  common  plane  being  unchanged,  we  have 


The  axis  of  y  may  be  regarded  as  an  axis  of  revolution  about 
which  the  planes  of  yx  and  yz  revolve  in  passing  from  the  first  to 
the  second  system  ;  and  if  u  denotes  the  angular  measure  of  this 
revolution,  or  u  =  XOX^  =  ZOZl  =  BAB»  we  readily  derive 
from  the  figure  the  equation 

x  sec  u  =  x^  —  zt  tan  u 
or,  multiplying  by  cos  u, 

x  =  XL  cos  u  —  zv  sin  u 
and 

z  =  x  tan  u  -{-  zl  sec  u 

or,  substituting  in  this  the  preceding  value  of  or, 
z  =x±  sin  u  -j-  z^  cos  u 

Thus,  to  pass  from  the  first  to  the  second  system,  we  have  the 
formulae 

x  =  xl  cos  M  —  2j  sin  u  *\ 

y=yi  >   (44) 

2  =  x±  sin  u  -f-  z±  cos  u  ) 

And  to  pass  from  the  second  to  the  first,  we  obtain  with  the  same 
ease, 

xt  ='     x  cos  u  -f  z  sin  u  ^ 

y.=    v  >  (45) 

zt  =  —  x  sin  u  -f-  £  cos  u  ) 

As  an  example,  let  us  apply  these  to  transforming  from  our 
second  system  of  spherical  co-ordinates  to  the  first ;  that  is,  from 
declination  and  hour  angle  to  altitude  and  azimuth.  The  meri- 
dian is  the  common  plane  ;  the  axis  of  z  in  the  system  of  declina- 
tion and  hour  angle  is  the  axis  of  the  equator,  and  the  axis  of  zl 
in  the  system  of  altitude  and  azimuth  is  the  vertical  line ;  the 
angle  between  these  axes  is  the  complement  of  the  latitude,  or 
u  =  90°  —  (p.  Substituting  this  value  of  u  in  (44),  and  also  the 
values  of  x,  y,  z,  xv  yv  zv  given  by  (39)  and  (40),  we  have,  after 
omitting  the  common  factor  J, 

VOL.  I.— 4 


50  THE    CELESTIAL    SPHERE. 

cos  h  cos  A  =  sin  y  cos  d  cos  t  —  cos  y  sin  d 
cos  h  sin  J.  =  cos  d  sin  t 

sin  A  =  cos  <f>  cos  £  cos  t  -f-  sin  ^  sin  5 

which  agree  with  (14).  We  see  that  when  the  element  of  dis- 
tance is  left  out  of  view  (as  it  must  necessarily  be  when  the 
origin  is  not  changed),  the  transformation  by  means  of  rectangu- 
lar co-ordinates  leads  to  the  same  forms  as  the  direct  application 
of  Spherical  Trigonometry.  With  regard  to  the  entire  generality 
of  these  formulae  in  their  application  to  angles  of  all  possible 
magnitudes,  see  Sph.  Trig.  Chap.  IV. 

DIFFERENTIAL   VARIATIONS    OF   CO-ORDINATES. 

34.  It  is  often  necessary  in  practical  astronomy  to  determine 
what  effect  given  variations  of  the  data  will  produce  in  the  quan- 
tities computed  from  them.  Where  the  formulae  of  computa- 
tion are  derived  directly  from  a  spherical  triangle,  we  can  employ 
for  this  purpose  the  equations  of  finite  differences  [Sph.  Trig. 
Chap.  VI.]  if  we  wish  to  obtain  rigorously  exact  relations,  or 
the  simpler  differential  equations  if  the  variations  considered 
are  extremely  small.  As  the  latter  case  is  very  frequent,  I  shall 
deduce  here  the  most  useful  differential  formulae,  assuming  as 
well  known  the  fundamental  ones  [Sph.  Trig.  Art.  153], 

da  —  cos  C  db  —  cos  B  dc  =  sin  b  sin  C  dA  ^ 

—  cos  C  da  -f-  db  —  cos  A  dc  —  sin  c  sin  A  dB  V     (46) 

—  cos  B  da  —  cos  A  db  -f-  dc  —  sin  a  sin  B  dC  ) 

From  these  we  obtain  the  following  by  eliminating  da: 

sin  C  db  —  cos  a  sin  B  dc  =  sin  b  cos  C  dA  -f-  sin  a  dB    )  ,^ 
—  cos  a  sin  C  db  -f-  sin  B  dc  —  sin  c  cos  B  dA  -)-  sin  a  dC     j 

and  by  eliminating  db  from  these : 

sin  a  sin  B  dc  —  cos  b  dA  -\-  cos  a  dR  -f  dC  (48) 

If  we  eliminate  dA  from  (47),  we  find 
cos  b  sin  C  db  —  cos  c  sin  B  dc  =  sin  c  cos  B  dB  —  sin  b  cos  C  dC 

the  terms  of  which  being  divided  either  by  sin  b  sin  C,  or  by  its 
equivalent  sin  c  sin  B,  we  obtain 

cot  b  db  —  cot  c  dc  =  cot  B  dB  —  cot  C  dC  (49) 


DIFFERENTIALS    OP   CO-ORDINATES.  51 

35.  As  an  example,  take  the  spherical  triangle  formed  by  the 
zenith,  the  pole,  and  a  star,  Art.  10,  and  put 

A  =  180°  —  A  a  =  9Q°  —  3 


then  the  first  equations  of  (46)  and  (47)  give 

d8  =  —  cos  q  dZ  -f-  sin  q  sin  C  dA  -}-  cos  t  d<p    )      ,c  AN 

cos  d  dt  =       sin  q  dZ  -f-  cos  q  sin  C  dA  -j-  sin  £  sin  t  dtp    } 

which  determine  the  errors  dd  and  dt  in  the  values  of  d  and  t 
computed  according  to  the  formulae  (4),  (5),  and  (6),  when  f,  A, 
and  <p  are  affected  by  the  small  errors  d£,  dA,  and  d<p  respectively. 
In  a  similar  manner  we  obtain 

d*  =  —  cos  q  dd  -f-  sin  q  cos  d  dt  -f-  cos  A  d<p    ")      ..  *  x 

8inC<^4.—        sin  #  dd  -J-  cos  <?  cos  3  dt  —  cosCsinJ.^    j 

which  determine  the  errors  d£  and  dA  in  the  values  of  f  and  ^1 
computed  by  (14),  when  £,  /,  and  <p  are  affected  by  the  small 
errors  d8,  dt,  and  d<p  respectively. 

36.  As  a  second  example,  take  the  triangle  formed  by  the  pole 
of  the  equator,  the  pole  of  the  ecliptic,  and  a  star,  Art.  23.  De- 
noting the  angle  at  the  star  by  57,  we  find 

dft  =  cos  TI  dd  —  sin  17  cos  8  da  -  sin  A  ds        \      .-2 

cos  p  dX  =  sin  i?  dd  -f  cos  77  cos  <5  rfa  -f-  sin  p  cos  /I  ck        j 

and  reciprocally, 

dd  =       cos  TJ  dft  -f  sin  y  cos  /9  <£A  -f-  sin  a  ds    )     ,.QN 

cos  d  da  =  —  sin  y£p.--\-.  cos  9  cos  fi  dX  —  sin  d  cos  ade    ) 


52  TIME. 


CHAPTER   II. 

TIME USE   OF   THE   EPHEMERIS — INTERPOLATION — STAR 

CATALOGUES. 

37.  TRANSIT. — The  instant  when  any  point  of    the  celestial 
sphere  is  on  the  meridian  of  an  observer  is  designated  as  the 
transit  of  that  point  over  the  meridian ;  also  the  meridian  passage, 
and   culmination.      In    one   complete   revolution  of  the   sphere 
about  its  axis,  every  point  of  it  is  twice  on  the  meridian,  at 
points  which  are  180°  distant  in  right  ascension.     It  is  therefore 
necessary  to  distinguish  between  the  two  transits.     The  meri- 
dian is  bisected  at  the  poles  of  the  equator:  the  transit  over  that 
half  of  the  meridian  which  contains  the  observer's  zenith  is  the 
upper  transit,  or  culmination;    that  over  the  half  of  the  meri- 
dian which  contains  the  nadir  is  the  lower  transit,  or  culmina- 
tion.    At  the  upper  transit  of  a  point  its  hour  angle  is  zero,  or 
07t ;  at  the  lower  transit,  its  hour  angle  is  12*. 

38.  The  motion  of  the  earth  about  its  axis  is  perfectly  uni- 
form.     If,  then,  the  axis  of  the  earth  preserved  precisely  the 
same  direction  in  space,  the   apparent  diurnal  motion  of  the 
celestial  sphere  would  also  be  perfectly  uniform,  and  the  inter- 
vals between  the  successive  transits  of  any  assumed  point  of  the 
sphere  would  be  perfectly  equal.     The  effect  of  changes  in  the 
position  of  the  earth's  axis  upon  the  transit  of  stars  is  most  per- 
ceptible in  the  case  of  stars  near  the  vanishing  points  of  the 
axis,  that  is,  near  the  poles  of  the  heavens.    We  obtain  a  measure 
of  time  sensibly  uniform  by  employing  the  successive  transits  of 
a  point  of  the  equator.     The  point  most  naturally  indicated  is 
the  vernal  equinox  (also  called  the  First  point  of  Aries,  and  de- 
noted by  the  symbol  for  Aries,  T). 

39.  A  sidereal  day  is  the  interval  of  time  between  two  succes- 
sive (iipper)  transits  of  the  true  vernal  equinox  over  the  same 
meridian. 

The  effect  of  precession  and  nutation  upon  the  time  of  transit 


TIME.  53 

of  the  vernal  equinox  is  so  nearly  the  same  at  two  successive 
transits,  that  sidereal  clays  thus  defined  are  sensibly  equal.  (See 
Chapter  XL  Art.  411.) 

The  sidereal  time  at  any  instant  is  the  h'our  angle  of  the  vernal 
equinox  at  that  instant,  reckoned  from  the  meridian  westward 
from  0*  to  24\ 

When  <Y>  is  on  the  meridian,  the  sidereal  time  is  0*  Om  0* ;  and 
this  instant  is  sometimes  called  sidereal  noon. 

40.  A  solar  day  is  the  interval  of  time  between  two  successive 
upper  transits  of  the  sun  over  the  same  meridian. 

The  solar  time  at  any  instant  is  the  hour  angle  of  the  sun  at 
that  instant. 

In  consequence  of  the  earth's  motion  about  the  sun  from  west 
to  east,  the  sun  appears  to  have  a  like  motion  among  the  stars, 
or  to  be  constantly  increasing  its  right  ascension ;  and  hence  a 
solar  day  is  longer  than  a  sidereal  day. 

41.  Apparent  and  mean  solar  time. — If  the  sun  changed  its  right 
ascension  uniformly,  solar  days,  though  not  equal  to  sidereal  days, 
would  still  be  equal  to  each  other.    But  the  sun's  motion  in  right 
ascension  is  not  uniform,  and  this  for  two  reasons  : 

1st.  The  sun  does  not  move  in  the  equator,  but  in  the  ecliptic, 
so  that,  even  were  the  sun's  motion  in  the  ecliptic  uniform,  its 
equal  changes  of  longitude  would  not  produce  equal  changes  of 
right  ascension;  2d.  The  sun's  motion  in  the  ecliptic  is  not  uni- 
form. 

To  obtain  a  uniform  measure  of  time  depending  on  the  sun's 
motion,  the  following  method  is  adopted.  A  fictitious  sun,  which 
we  shall  call  the^r,^  mean  sun,  is  supposed  to  move  uniformly  at 
such  a  rate  as  to  return  to  the  perigee  at  the  same  time  with  the 
true  sun.  Another  fictitious  sun,  which  we  shall  call  the  second 
mean  sun  (and  which  is  often  called  simply  the  mean  sun),  is  sup- 
posed to  move  uniformly  in  the  equator  at  the  same  rate  as  the 
first  mean  sun  in  the  ecliptic,  and  to  return  to  the  vernal  equinox 
at  the  same  time  with  it.  Then  the  time  denoted  by  this  second 
mean  sun  is  perfectly  uniform  in  its  increase,  and  is  called  mean  time. 

The  time  which  is  denoted  by  the  true  sun  is  called  the  true 
or,  more  commonly,  the  apparent  time. 

The  instant  of  transit  of  the  true  sun  is  called  apparent  noon,  and 
the  instant  of  transit  of  the  second  mean  sun  is  called  mean  noon. 


54  TIME. 

The  equation  of  time  is  the  difference  between  apparent  and 
mean  time ;  or,  in  other  words,  it  is  the  difference  between  the 
hour  angles  of  the  true  sun  and  the  second  mean  sun.  The 
greatest  difference  is  about  16™ 

The  equation  of  time  is  also  the  difference  between  the  right 
ascensions  of  the  true  sun  and  the  second  mean  sun.  The  right 
ascension  of  the  second  mean  sun  is,  according  to  the  preceding 
definitions,  equal  to  the  longitude  of  the  first  mean  sun,  or,  as  it 
is  commonly  called,  the  sun's  mean  longitude.  To  compute  the 
equation  of  time,  therefore,  we  must  know  how  to  find  the  longi- 
tude of  the  first  mean  sun ;  and  this  is  deduced  from  a  knowledge 
of  the  true  sun's  apparent  motion  in  the  ecliptic,  which  belongs 
to  Physical  Astronomy.  Here  it  suffices  us  that  its  value  is 
given  for  each  day  of  the  year  in  the  Ephemeris,  or  Nautical 
Almanac. 

42.  Astronomical  time. — The   solar  day  (apparent  or  mean)  is 
conceived  by  astronomers  to  commence  at  noon  (apparent  or 
mean),  and  is  divided  into  twenty-four  hours,  numbered  succes- 
sively from  0  to  24. 

Astronomical  time  (apparent  or  mean)  is,  then,  the  hour  angle 
of  the  sun  (apparent  or  mean),  reckoned  on  the  equator  west- 
ward throughout  its  entire  circumference  from  0*.to  24\ 

43.  Civil  time. — For  the  common  purposes  of  life,  it  is  more 
convenient  to  begin  the  day  at  midnight,  that  is,  when  the  sun 
is  on  the  meridian  at  its  lower  transit 

The  civil  day  is  divided  into  two  periods  of  twelve  hours  each, 
namely,  from  midnight  to  noon,  marked  A.M.  (Ante  Meridiem), 
and  from  noon  to  midnight,  marked  P.M.  (Post  Meridiem) 

44.  To  convert  civil  into  astronomical  time. — The  civil  day  begins 
12*  before  the  astronomical  day  of  the  same  date.  '  This  remark 
is  the  only  precept  that  need  be  given  for  the  conversion  of  one 
of  these  kinds  of  time  into  the  other. 

EXAMPLES. 

Ast.  T.  May  10.  15*=  Civ.  T.  May  11,  3*  A.M. 
Ast.  T.  Jan.  3,  7ft=  Civ.  T.  Jan.  3,  7*  P.M. 
Ast,  T.  Aug.  31,  20*  =  Civ.  T.  Sept.  1,  8*  A.M. 


TIME.  55 

45.   Time  at  different  meridians.  —  The  hour  angle  of  the  sun  at 
any  meridian  is  called  the  local  (solar)  time  at  that  meridian. 
The  hour  angle  of  the  sun  at  the  Greenwich  meridian  at  the 

O 

same  instant  is  the  corresponding  Greenwich  time.  This  time  we 
shall  have  constant  occasion  to  use,  both  because  longitudes 
in  this  country  and  England  are  reckoned  from  the  Greenwich 
meridian,  and  because  the  American  and  British  Nautical 
Almanacs  are  computed  for  Greemvich  time.* 

The  difference  between  the  local  time  at  any  meridian  and  the 
Greenwich  time  is  equal  to  the  longitude  of  that  meridian  from 
Greenwich,  expressed  in  time,  observing  that  1*  =  15°. 

The  difference  between  the  local  times  of  any  two 
meridians  is  equal  to  the  difference  of  longitude  of 
those  meridians. 

In  comparing  the  corresponding  times  at  two  dif- 
ferent meridians,  the  most  easterly  meridian  may  be 
distinguished  as  that  at  which  the  time  is  greatest; 

4.1.     .•       i     . 

that  is,  latest. 

If  then  PMj  Fig.  10,  is  any  meridian  (referred  to  the  celestial 
sphere),  PG  the  Greenwich  meridian,  PS  the  declination  circle 
through  the  sun,  and  if  we  put 

T0  =  the  Greenwich  time  =  GPS, 

T  =  the  local  time  =  MPS, 

L  =  the  west  longitude  of  the  meridian  PM  =  GPM, 

we  have 

y    ___  rrt  rri 

° 


If  the  given  meridian  were  east  of  Greenwich,  as  PM',  we 
should  have  its  east  longitude  =  T  —  T0;  but  we  prefer  to  use 
the  general  formula  L  =  T0  —  T7  in  all  cases,  observing  that  east 
longitudes  are  to  be  regarded  as  negative. 

In  the  formula  (54),  TQ  and  T  are  supposed  to  be  reckoned 
always  westward  from  their  respective  meridians,  and  from  Qh  to 
24A  ;  that  is,  TQ  and  T  are  the  astronomical  time.s,  which  should,  of 
course,  be  used  in  all  astronomical  computations. 

As  in  almost  every  computation  of  practical  astronomy  we  are 
dependent  for  some  of  the  data  upon  the  ephemeris,  —  and  these 

*  What  we  have  to  say  respecting  the  Greenwich  time  is,  however,  equally  appli- 
cable to  the  time  at  any  other  meridian  for  which  the  ephemeris  may  be  computed. 


56  TIME. 

are  commonly  given  for  Greenwich, — it  is  generally  the  first  step 
in  such  a  computation  to  deduce  an  exact  or,  at  least,  an  ap- 
proximate value  of  the  Greenwich  astronomical  time.  It  need 
hardly  be  added  that  the  Greenwich  time  should  never  be  other- 
wise expressed  than  astronomically.* 

EXAMPLES. 

1.  In  Longitude  76°  32'  W.  the  local   time  is  1856  April  1, 
9*  3W  10s  A.M. ;  what  is  the  Greenwich  time  ? 

Local  Ast,  T.  March  31,    2P    3-    10s 

Longitude  -(-56         8 

Greenwich  T.  AprilTT^      2 9       18 

2.  In  Long.  105°  15'  E.  the  local  time  is  August  21,  4*  3m  P.M ; 
what  is  the  Greenwich  time  ? 

Local  Ast.  T.  Aug.  21,      4*    3™ 
Longitude  7      1 

Greenwich  T.  Aug.  20,    21      2 

3.  Long.  175°  30'  W.  Loc.  T.  Sept,  30,  Sh  I0m  A.M.  =  G.  T. 
Sept.  30,  V  52*. 

4.  Long.  165°  0'  E.  Loc.  T.  Feb.  1,  7*  llm  P.M.  =  G.  T.  Jan. 
31,  20*  II"1. 

5.  Long.  64°  30'  E.  Loc.  T.  June  1,  Oh  M.  (Noon)  =  G.  T.  May 
31,  17"  17"1  56s. 

46.  In  nautical  practice  the  observer  is  provided  with  a  chro- 
nometer which  is  regulated  to  Greenwich  time,  before  sailing, 
at  a  place  whose  longitude  is  well  kno\vn.  Its  error  on  Green- 
wich time  is  carefully  determined,  as  well  as  its  daily  gain  or 
loss,  that  is,  its  rate,  so  that  at  any  subsequent  time  the  Green- 
wich time  may  be  known  from  the  indication  of  the  chronometer 
corrected  for  its  error  and  the  accumulated  rate  since«the  date 
of  sailing.  As,  however,  the  chronometer  has  usually  only  12* 
marked  on  the  dial,  it  is  necessary  to  distinguish  whether  it 
indicates  A.M.  or  P.M.  at  Greenwich.  This  is  always  readily 
done  by  means  of  the  observer's  approximate  longitude  and  local 

*  On  this  account,  chronometers  intended  for  nautical  and  astronomical  purposes 
should  always  be  marked  from  C*  to  24A,  instead  of  from  0*  to  12*  as  is  now  usual. 
It  is  surprising  that  navigators  have  not  insisted  upon  this  point. 


TIME.  57 

time.     As  this  is  a  daily  operation  at  sea,  it  may  be  well  to  illus- 
trate it  by  a  few  examples. 

1.  In  the  approximate  longitude  5h  W.  about  3/l  P.M.  on  Au- 
gust 3,  the  Greenwich  Chronometer  marks  8A  llw  7",  and  is  fast 
on  G.  T.  6"*  10s ;  what  is  the  Greenwich  astronomical  time  ? 

Approx.  Local  T.  Aug.  3,  3*        Gr.  Chronom.  8*  llm    7' 

"         Longitude,       -f-  5          Correction,  —    6    10 

Approx.  G.  T.  Aug.  3,       8          Gr.  Ast.  T.  Aug.  3,  8      4    57 

2.  In  Long.  10*  E.  about  1*  A.M.  on  Dec.  7,  the  Greenwich 
Chronometer  marks  3*  14ni  13*.5,  and  is  fast  25™  18*.7 ;  what  is 
the  G.  T.  ? 

Approx.  Local  T.  Dec.  6,  13*  Gr.  Chronom.      3*  Um  13*.5 

"        Long.  - 10  Correction,  -  25    18  .7 

Approx.  G.  T.  Dec.  6,       T  G.  A.  T.  Dec.  6,  2  48    54.8 

3.  In  Long.  9"  12™  W.  about  2*  A.M.  on  Feb.  13,  the  Gr.  Chron. 
marks  10"  2?"  13*.3,  and  is  slow  30'"  30*.3;  what  is  the  G.  T.? 

Approx.  Local  T.  Feb.  12,  14*         Gr.  Chronom.        10*  37ro  13S.3 

"        Long.  -f9          Correction,  -f    30    30.3 

Approx.  G.  T.  Feb.  12,        23          G.  A.  T.  Feb.  12,  23     7    43  -6 

The  computation  of  the  approximate  Greenwich  time  may,  of 
course,  be  performed  mentally. 

47.  The  formula  (54),  L=  T0—  71,  is  true  not  only  when   T0 
and  T  are  solar  times,  but  also  when  they  are  any  kinds  of  time 
whatever,  or,  in  general,  when  jT0  and  T  express  the  hour  angles 
of  any  point  whatever  of  the  sphere  at  the  two  meridians  whose 
difference  of  longitude  is  L.    This  is  evident  from  Fig.  10,  where 
S  may  be  any  point  of  the  sphere. 

48.  To  convert  the  apparent  time  at  a  given  meridian  into  the  mean 
time,  or  the  mean  into  the  apparent  time. 

If  M  —  the  mean  time, 

A  =  the  corresponding  apparent  time 
E  =.  the  equation  of  time, 
we  have 

M  =  A  -f  E 
or  A  =M—E 


58  TIME. 

in  which  E  is  to  be  regarded  as  a  positive  quantity  when  it  is 
additive  to  apparent  time.  The  value  of  E  is  to  be  taken  from  the 
Nautical  Almanac  for  the  Greenwich  instant  corresponding  to 
the  given  local  time.  If  apparent  time  is  given,  find  the  Gr. 
apparent  time  and  take  E  from  page  I  of  the  month  in  the 
Nautical  Almanac;  if  mean  time  is  given,  find  the  Gr.  mean 
time  and  take  E  from  page  II  of  the  month. 

EXAMPLE  1.—  In  longitude  60°  W.,  1856  May  24,  3A  12W  10s 
P.M.,  apparent  time  ;  what  is  the  mean  time  ? 
We  have  first 

Local  time  May  21,        3"  12-  10' 

Longitude,  400 

Gr.  app.  time  May  24,    7    12    10 

We  must,  therefore,  find  E  for  the  Gr.  time,  May  24,  lh  12'* 
10',  or  7A.21.  By  the  Nautical  Almanac  for  1856,  we  have  Eat 
apparent  Greenwich  noon  May  24  =  —  3'"  25*.43,  and  the  hourly 
difference  -f  0*.224.  Hence  at  the  given  time 

E  =  —  3-  25'.43  +  0'.224  X  7.21  =  —  3-  23'.81 
and  the  required  mean  time  is 

M=  3A  12-  10-  —  3-  23«.81  =  3*  8-  46*.19. 

EXAMPLE  2.—  In  longitude  60°  W.,  1856  May  24,  37'  8W  46M9 
mean  time  ;  what  is  the  apparent  time  ? 

Gr.  mean  time,  May  24,  7*  8-  46M9  (=  7M5) 

E  at  mean  noon  May  24  =  -     3™  25*.41     Hourly  diff.  =  0'.224 

Correction  for  7M5  +    1.60  7.15 

E  =  —  3     23.81  T760 

and  hence 

M=S*  8-46'.19 

—  E  =  3    23  .81 


A  =  3    12    1(OO 

As  the  equation  of  time  is  not  a  uniformly  varying  quantity,  it 
is  not  quite  accurate  to  compute  its  correction  as  above,  by  mul- 
tiplying the  given  hourly  difference  by  the  number  of  hours  in 
the  Greenwich  time,  for  that  process  assumes  that  this  hourly 
difference  is  the  same  for  each  hour.  The  variations  in  the 
hourly  difference  are,  however,  so  small  that  it  is  only  when 


TIME.  59 

extreme  precision  is  required  that  recourse  must  be  had  to  the 
more  exact  method  of  interpolation  which  will  be  given  here- 
after. 

49.  To  determine  the  relative  length  of  the  solar  and  sidereal  units 
of  time. 

According  to  BESSEL,  the  length  of  the  tropical  year  (which  is 
the  interval  between  two  successive  passages  of  the  sun  through 
the  mean  vernal  equinox)  is  365.24222  mean  solar  days;*  and 
since  in  this  time  the  mean  sun  has  described  the  whole  arc  of 
the  equator  included  between  the  two  positions  of  the  equinox, 
it  has  made  one  transit  less  over  any  given  meridian  than  the 
vernal  equinox  ;  so  that  we  have 

366.24222  sidereal  days  =  365.24222  mean  solar  days 
whence  we  deduce 


. 

1  sid.  day  =  *IZz  sol.  day  =  0.99726957  sol.  day 
366.24222 

or 

24*  sid.  time  =  23*  56m  48,091  solar  time 
Also, 

Qfifi  91999 

1  sol.  day  =  -  -  sid.  day  =  1.00273791  sid.  day 

365.24222 

or 

24*  sol.  time  =  24*  3M  56'.555  sid.  time 

If  we  put 

_ 


365.24222 

and  denote  by  /an  interval  of  mean  solar  time,  by  If  the  equiva- 
lent interval  of  sidereal  time,  we  always  have 

I'  =  fJJ  =  1  +(11—  1)1   =1+  .00273791  I 


I  =  ~  ==l'  —  ri  —  ±)I'  =  r—  .00273043/'        f      (55) 
/"  v  ) 

Tables  are  given  in  the  Nautical  Almanacs  to  save  the  labor  of 
computing  these  equations.  In  some  of  these  tables,  for  each 
solar  interval  /  there  is  given  the  equivalent  sidereal  interval 
I'  =  pi,  and  reciprocally  :  in  others  there  are  given  the  correc- 
tion to  be  added  to  /to  find  I'  (i.e.  the  correction  .00273791  /), 


*  The  length  of  the  tropical  year  is  not  absolutely  constant.     The  value  given  in 
the  text  is  for  the  year  1800.     Its  decrease  in  100  years  is  about  C*.6  (Art.  407). 


60  TIME. 

and  the  correction  to  be  subtracted  from  I1  to  find  /  (i.e.  the 
correction  .00273043  I').  The  latter  form  is  the  most  conve- 
nient, and  is  adopted  in  the  American  Ephemeris.  The  correction 
(p.  —  1)  /  is  frequently  called  the  acceleration  of  the  fixed  stars  (re- 
latively to  the  sun).  The  daily  acceleration  is  3W  56\555. 

50.  To  convert  the  mean  solar  time  at  a  given  meridian  into  the 
corresponding  sidereal  time. 

In  Fig.  1,  page  25,  if  PQ  is  the  given  meridian,  VQ  the  equator, 
D  the  mean  sun,  Vthe  vernal  equinox,  and  if  we  put 

T  —  DQ  =  the  mean  solar  time, 
0  =  VQ  =  the  sidereal  time, 

=  the  right  ascension  of  the  meridian, 
V==  the  right  ascension  of  the  mean  sun, 
we  have 

0=  T+  V  (56) 


The  right  ascension  of  the  mean  sun,  or  V,  is  given  in  the 
American  Ephemeris,  on  page  II  of  the  month,  for  each  Green- 
wich mean  noon.  It  is,  however,  there  called  the  "  Sidereal 
Time,"  because  at  mean  noon  the  second  mean  sun  is  on  the 
meridian,  and  its  right  ascension  is  also  the  right  ascension  of 
the  meridian,  or  the  sidereal  time.  But  this  quantity  V  is  uni- 
formly increasing*  at  the  rate  of  3m  56*.555  in  24  mean  solar 
hours,  or  of  9*.8565  in  one  mean  hour.  To  find  its  value  at  the 
given  time  T,  we  may  first  find  the  Greenwich  mean  time  TQ  by 
applying  the  longitude  ;  then,  if  we  put 

V0  —  the  value  of  V  &t  Gr.  mean  noon, 

=  the  "  sidereal  time"  in  the  ephemeris  for  the  given  date, 

we  have 

V=  V0  +  9-.85G5  X  T0 

in  which  TQ  must  be  expressed  in  hours  and  decimal  parts.  It 
is  easily  seen  that  9".8565  is  the  acceleration  of  sidereal  time  on 
solar  time  in  one  solar  hour,  and  therefore  the  term  9*.  8565  X  T0 
is  the  correction  to  add  to  TQ  to  reduce  it  from  a  solar  to  a  side- 
real interval.  This  term  is  identical  with  (//  —  1)!T0  as  given  by 


*  The  sidereal  time  at  mean  noon  is  equal  to  the  true  R.A.  of  the  mean  sun,  or  it 
is  the  R.A.  of  the  mean  sun  referred  to  the  true  equinox,  and  therefore  involves  the 
nutation,  so  that  its  rate  of  increase  is  not,  strictly,  uniform.  But  it  is  sufficiently  so 
for  24  hours  to  be  so  regarded  in  all  practical  computations.  See  Chapter  XI. 


TIME.  61 

the  preceding  article,  if  7^  in  the  latter  expression  is f expressed 
in  seconds,  since  we  have 


We  may  then  write  (56)  in  the  following  form,  putting  L  =  the 
west  longitude  of  the  given  meridian,  and  T0=  T -\-  L: 

e=  T  +  F0  +  (A* -  1)  (T+  L)  (57) 

The  term  (JJL  —  1)  (T  -\-  L)  is  given  in  the  tables  of  the  Amer- 
ican Ephemeris  for  converting  "Mean  into  Sidereal  Time,"  and 
may  be  found  by  entering  the  table  with  the  argument  T  -\-  L, 
or  by  entering  successively  with  the  arguments  T  and  L  and 
adding  the  corrections  found,  observing  to  give  the  correction 
for  the  longitude  the  negative  sign  when  the  longitude  is  east. 
If  no  tables  are  at  hand,  the  direct  computation  of  this  term  will 
be  more  convenient  under  the  form  9s. 8565  X  TQ. 

EXAMPLE  1.— In  Longitude  165°  W.  1856  May  17,  4*  A.M.; 
what  is  the  sidereal  time  ? 

The  Greenwich  time  is  May  17,  3A;  and  the  computation  may 
be  arranged  as  follows : 

Local  Ast.  Time  T  =  16*    Om  0'. 

At  Gr.  Noon  May  17,        V0=    3  41  28  .32 
Correction  of  Fft  for  3* 


29  i 
=  9'.8565  X  3       ' 


=  19   41  57.89 


EXAMPLE  2.— In  Longitude  25°  17'  E.  1856  March  13,  about 
9''  30'*  P.M.,  an  observation  is  noted  by  a  Greenwich  chronometer 
which  gives  7/l  51W  12*.3  and  is  slow  3'n  13*.4 ;  what  is  the  local 
sidereal  time  ? 

Gr.  mean  date,  March  13,  7*  54"  25'.7 

Longitude,  1  41      8      E. 

T  =   9  35    33.7 

March  13,  F0  =  23  25    12.26 
Tabular  corr.  for  7*  54"  25'.7  =  1    17  .94 

8=~9      2     3^90 


62  TIME. 

EXAMPLE  3.— In  Longitude  7;'  25-  12<  E.  1856  March  13, 13*  15™ 
47*. 3  mean  local  astronomical  time  ;  what  is  the  sidereal  time  ? 

T  =13*15"147S.3 
F0=  23    25  12  .26 

Tabular  corr.  for          13*  15*  47*.3  '  =   -f     2  10  .73 
Tab.  corr.  for  long.  —  7*  25m  12».     ==~  '-        1  13  .14 

0^=  12   41  57.15 

51.  To  convert  the  apparent  solar  time  at  a  given  meridian  into  the 
sid,ercal  time  at  that  meridian. 

Find  the  mean  time  by  Art.  48,  and  then  the  sidereal  time  by 
Art.  50. 

Or,  more  directly,  to  the  given  apparent  time  add  the  true  sun's 
right  ascension.  For  if  in  Fig.  1  we  take  D  as  the  true  sun,  we 
have  DQ  =  apparent  solar  time,  VD  =  R.  A.  of  true  sun,  and 
VQ,  the  sidereal  time,  is  the  sum  of  these  two. 

The  right  ascension  of  the  true  sun  is  called  in  the  Ephemeris 
the  "  sun's  apparent  right  ascension,"  and  is  there  given  for  each 
apparent  noon.  It  is  not  a  uniformly  increasing  quantity;  but 
for  many  purposes  it  will  be  sufficiently  accurate  to  consider  the 
hourly  increase  given  in  the  Ephemeris  as  constant  for  24\  and 
to  add  to  the  app.  R.  A.  of  the  Ephemeris  the  correction  found 
by  multiplying  the  hourly  difference  by  the  number  of  hours  in 
the  Greenwich  time. 

EXAMPLE.— In  Longitude  98°  W.  1856  June  3,  4*  10"'  P.M. 
app.  time  ;  what  is  the  sidereal  time  ? 

Gr.  app.  date  June  3,  10*  42™  (=  10*.7)  Local  app.  t.  =  4*  10TO  0'. 

O's  App.  R.  A,  App.  noon  June  3    =  4    46  22  .04 
Hourly  diff.  =  10«.271     Corr.  =  10'.271  X  10.7          =          1  49.90 

Sidereal  time  =8    58  11 .94 

52.  To  convert  the  sidereal  time  at  a  given  meridian  into  the  mean 
time  at  that  meridian. 

First  method. — When  the  Greenwich  mean  time  is  also  given, 
as  is  frequently  the  case,  we  have  only  to  find  V  as  in  Art.  50 
by  adding  to  V0  given  in  the  Ephemeris  the  correction  for  the 
Greenwich  time  taken  from  the  table  "Mean  into  Sidereal 
Time,"  and  then  we  have,  by  transposing  equation  (56), 

T=  0  —  V 


TIME.  63 

EXAMPLE.— In  Longitude  165°  W.,  the  Greenwich  mean  time 
being  1856  May  17,  3*,  the  local  sidereal  time  19*  41*  57*.89, 
what  is  the  local  mean  time  ? 

=    3*  41-  28«.32 


»    0 

Corr.  for  3* 

V 
0 

0  —  F  =  T 

— 

Tt  JL 

+ 

IdU    .«J^ 

29.57 

=    3 
=  19 

41 
41 

57.89 
57  .89 

=  16 

0 

0.00 

The  longitude  being  H7t  "W.,  the  local  date  is  May  16. 

Second  method.  —  When  the  Greenwich  mean  time  is  not  given, 
we  can  find  T  from  (57),  all  the  other  quantities  in  that  equation 
being  known.  We  find 


I1- 


or,  in  a  more  convenient  form  for  use, 

T  =  0  -  F0  -     l  -    ~     (0  -F0  +  £)  (58) 


in  which  the  term  multiplied  by  1  —  —  is  the  retardation  of  mean 

time  on  sidereal  in  the  interval  0  —  F0  +  L,  and  is  given  in  the 
table  "Sidereal  into  Mean  Time."  It  is  convenient  to  enter  the 
table  first  with  the  argument  0  --  F0  and  then  with  the  argu- 
ment L,  and  to  subtract  the  two  corrections  from  0  —  F0,  ob- 
serving that  the  correction  for  the  longitude  becomes  additive 
if  the  longitude  is  east. 

EXAMPLE.  —  In  Longitude  165°  W.  1856  May  16,  the  sidereal 
time  is  19/l  41W  57*.89  ;  what  is  the*  mean  local  time? 

0  =  19*  41*  57'.89 

May  16,     F0  =    3   37    31.76 

0  —  F0  =  16     4    26  .13 

Table,  "Sidereal  into  j  Corr.  for  16-'  4m  26'.13       =       -  2    38  .00 
Mean  Time"  I      "       "  longitude    11*      =_    -  1    48  .13 

T=~IQ     0      0^00 

53.  The  following  method  of  converting  the  sidereal  into  the 

mean  time  is  preferred  by  some.     In  the  last  column  of  page  III 

of  the  month  in  the  American  Naut.  Aim.  is  given  the  "Mean 

Time  of  Sidereal  0V     This  quantity,  which  we  may  denote  by 

F',  is  the  number  of  hours  the  mean  sun  is  west  of  the  vernal 


64  TIME. 

equinox,  and  is  merely  the  difference  between  24A  and  the  mean 
sun's  right  ascension.  The  hour  angle  of  the  mean  sun  at  any 
instant  is  then  the  hour  angle  of  the  vernal  equinox  increased 
by  the  value  of  V  at  that  instant.  To  find  this  value  of  V1 ',  we 
first  reduce  the  Almanac  value  to  the  given  meridian  by  cor- 
recting it  for  the  longitude  by  the  table  for  converting  sidereal 
into  mean  time ;  then  reduce  it  to  the  given  sidereal  time  0 
(which  is  the  elapsed  sidereal  time  since  the  transit  of  the  vernal 
equinox  over  the  given  meridian)  by  further  correcting  it  by  the 
same  table  for  this  time  0.  We  then  have  the  mean  time  7"by 
the  formula 

T=e  +  v 

It  is  necessary  to  observe,  however,  that  if  0  +  V  exceed 
24*  it  will  increase  our  date  by  one  day;  and  in  that  case  V 
should  be  taken  from  the  Almanac  for  a  date  one  day  less  than 
the  given  date;  that  is,  we  must  in  every  case  take  that  value 
which  belongs  to  the  Greenwich  transit  of  the  vernal  equinox 
immediately  preceding  that  over  the  given  meridian. 

EXAMPLE. — Same  as  in  Art.  52. 

0  =  19*  41-  57'.89 

May  15,  F0'  =  20    23      3  .88 

Corr.  for  long.  11*  W.  =        -  1    48  .13 

Corr.  for    19*  41-  58«  =  _    -3    13.64 

T  =  ~IQ      0      OTOO 

• 

54.  To  find  the  hour  angle  of  a  star*  at  a  given  time  at  a  given 
meridian. 

In  Fig.  1,  we  have  for  the  star  at  0,  DQ  =  VQ  —  FD;  that 
is,  if  we  put 

0  =  the  sidereal  time, 

a  =  the  right  ascension  of  the  star/ 

t  =  the  hour  angle  "     "      " 

then  t  =  0  —  «  .  (59) 

If  a  exceeds  0,  this  formula  will  give  a  negative  value  of  t 
which  will  express  the  hour  angle  east  of  the  meridian:  in  that 
case,  if  we  increase  0  by  24/l  before  subtracting  a,  we  shall  find 

*  We  shall  use  "star,"  for  brevity,  to  denote  any  celestial  body. 


HOUR    ANGLES.  65 

the  value  of  t  reckoned  in  the  usual  manner,  west  of  the  meri- 
dian. 

According  to  this  formula,  then,  we  have  first  to  convert  the 
given  time  into  the  sidereal  time,  from  which  we  then  subtract 
the  right  ascension  of  the  star,  increasing  the  sidereal  time  by 
24/l  when  necessary ;  the  remainder  is  the  required  hour  angle 
west  of  the  meridian. 

In  the  case  of  the  sun,  however,  the  apparent  time  is  at  once 
the  required  hour  angle,  and  we  only  have  to  apply  to  the  given 
mean  time  the  equation  of  time. 

EXAMPLE.— In  Longitude  165°  W.  1856  May  16, 16*  Om  0s  mean 
time,  find  the  hour  angles  of  the  sun,  the  moon,  Jupiter,  and 
the  star  Fomalhaut. 

The  Greenwich  mean  date  is  1856  May  17,  3*,  and  the  local 
sidereal  time  is  (see  Example  1,  Art.  50)  0  =  19*  41'*  57'.89. 
For  the  Greenwich  date  we  find  from  the  Naut.  Aim.  the  equa- 
tion of  time  E,  and  the  right  ascensions  a  of  the  moon,  Jupiter, 
and  Fomalhaut,  as  below : 

T=IG*    0"     0"  0  =  19*  41TO  57'.89 

-  E  =  -\-     3     49  .85  D's  a  =  13    50    21.35 

O'sf  =  16     3     49.85  D's  t  =  '5    51    36.54 

6  ==  19*  41*  57'.89  0  =  19*  41-  57*.89 

QJ/S  a  =    0      7     57  .52         Fomalh.  a  =  22    49    40  .18 


QJ>  t  =  19    34       0  .37         Fomalh.  t  =  20    52    17  .71 

If  the  sidereal  time  had  been  given  at  first,  we  should  have- 
found  the  hour  angle  of  the  sun  by  subtracting  its  apparent  right 
ascension  as  in  the  case  of  any  other  body. 

55.  Given  the  hour  angle  of  a  star  at  a  given  meridian  on  a  given 
day,  to  find  the  local  mean  time. 

By  transposing  the  formula  (59),  we  have 

&  =  t  -f  a  (60) 

so  that,  the  right  ascension  of  the  star  being  given,  we  have  only 
to  add  it  to  the  given  hour  angle  to  obtain  the  local  sidereal  time, 
whence  the  mean  time  is  found  by  Art.  52.  When  the  sum  t  -f-  a 
exceeds  24*,  we  must,  of  course,  deduct  24*.  If  the  body  is  the 
sun,  however,  the  given  hour  angle  is  at  once  the  apparent  time, 
whence  the  mean  time  as  before.  But  if  the  body  is  the  moon 

VOL.  I.— 5 


66  TIME. 

or  a  planet,  its  right  ascension  can  be  found  from  the  Ephemeris 
only  when  we  know  the  Greenwich  time.  If  then  the  Green- 
wich time  is  not  given,  we  must  find  an  approximate  value  of 
the  local  time  by  formula  (60),  using  for  a  a  value  taken  for  a 
Greenwich  time  as  nearly  estimated  as  possible ;  from  this  local 
time  deduce  a  more  exact  value  of  the  Greenwich  time,  with 
which  a  more  exact  value  of  a  may  be  found;  and  so  repeating  as 
often  as  maybe  necessary  to  reach  the  required  degree  of  precision. 

EXAMPLE  1.— In  Longitude  165°  W.  1856  May  16,  the  hour  angle 
of  Fomalhaut  is  207t  52Hl  17'.71;  what  is  the  mean  time  ? 

t  =  20*  52™  17'.71 

May  16,  Fomalh.  «  =  22  49    40.18 
6  =  19  41    57  .89 
whence  the  mean  time  is  found  to  be  T=  167t  Om  0*. 

EXAMPLE  2.— In  Longitude  165°  W.  1856  May  16,  the  moon's 
hour  angle  is  57'  51wl  36s. 54,  and  the  Greenwich  date  is  given  May 
17,  3* ;  what  is  the  mean  time  ? 


For  May  17,  3», 

"   May  17,  3ft, 

a  =  13  50  21  .35 

O  =  19  41  57  .89 
V=  3  41  57.89 

T  =  16     0      0  .00 

EXAMPLE  3. — In  Longitude  30°  E.  1856  August  10,  the  moon's 
hour  angle  is  4A  10"1  53".2;  what  is  the  mean  time  ? 

For  a  first  approximation,  we  observe  that  the  moon  passes  the 
meridian  on  August  10  at  about  lh  mean  time  (Am.  Eph.  page 
IV  of  the  month),  and  when  it  is  west  of  the  meridian  4fe  the 
mean  time  is  about  4*  later,  or  H7i,  from  which  subtracting  the 
longitude  2*  we  have,  as  a  rough  value  of  the  Greenwich  time 
Aug.  10,  9*.  "We  then  have 

t  =    4*  11"* 

For  Aug.  10,  9*,      «  =  16    29 
0  =  20    40 

"   Aug.  10,  9*,     V=    9    18 
1st  approx.  value     T  =  11    22 

Hence  the  more  exact  Greenwich  date  is  Aug.  10,  9*  22™;  and 
with  this  we  now  repeat: 


HOUR    ANGLES.  67 

t  =    4*  10~  53«.2 

For  Aug.  10,  9ft  22"     a  =  16   29    26.8 

0  =  20   40   20.0 

"  «  V=    9    18     8.1 

2d  approx.  value       T  =  11    22    11.9 

A  third  approximation,  setting  out  from  this  value  of  71,  gives 
us  T=  llh  22*  128.32. 

56.  The  mean  time  of  the  meridian  passage  not  only  of  the 
moon  but  of  each  of  the  planets  is  given  in  the  Ephemeris. 
This  quantity  is  nothing  more  than  the  arc  of  the  equator  in- 
tercepted between  the  mean  sun  and  the  moon's  or  planet's 
declination  circle.  If  we  denote  it  by  M9  we  may  regard  M  as 
the  equation  between  mean  time  and  the  lunar  or  planetary  time, 
these  terms  being  used  instead  of  "hour  angle  of  the  moon"  or 
"hour  angle  of  a  planet,"  just  as  we  use  "solar  time"  to  signify 
"hour  angle  of  the  sun."  This  quantity  Mis  given  in  the  Ephe- 
meris for  the  instant  when  the  lunar  or  planetary  time  is  0*,  and 
its  variation  in  1A  of  such  time  is  also  given  in  the  adjacent 
column.  If,  then,  when  the  moon's  or  a  planet's  hour  angle  at  a 
given  meridian  =  t,  we  take  out  from  the  Almanac  the  value  of 
M  for  the  corresponding  Greenwich  value  of  t,  we  shall  find  the 
mean  time  T  by  simply  adding  M  to  t ;  that  is, 

T=t+M  (61) 

This  is,  in  fact,  the  direct  solution  of  the  problem  of  the  pre- 
ceding article,  and  neither  requires  a  previous  knowledge  of  the 
Greenwich  mean  time  nor  introduces  the  sidereal  time.  But 
the  Almanac  values  of  M  are  not  given  to  seconds;  and  there- 
fore we  can  use  (61)  only  for  making  our  first  approximation  to 
jP,  after  which  we  proceed  as  in  the  last  article.  The  Green- 
wich value  of  t  with  which  we  take  out  M  is  equal  to  t  -f  L, 
denoting  by  L  the  longitude  of  the  given  meridian  (to  be  taken 
with  the  negative  sign  when  east),  and  the  required  value  of  M  is 
the  Almanac  value  increased  by  the  hourly  diff.  multiplied  by 
(t  +  L)  in  hours.  As  the  hourly  diff.  of  M  in  the  case  of  the  moon 
is  itself  variable,  we  should  use  that  value  of  it  which  corresponds 
to  the  middle  of  the  interval  t  -f-  L;  that  is,  we  should  first  correct 
the  hourly  diff.  by  the  product  of  its  hourly  change  into  J  (t  -f  Z/) 
in  hours. 


68  EPHEMERIS. 

EXAMPLE. — Same  as  Example  3,  Art.  55.     We  have 

t  -f  L  =  2*  10-  53'.2  =  2M8  t  =   4A  10"1  53'.2 

AtGr. trans.  Hour.  Diff.  =2m.l7  At Gr. trans. Aug.lO,3/=    7     6   30 
Variation  of  H.D.  in  1*  5"*  =     .01      2TO.18  X  2.18   '  =   -f-    4  45 

Corrected  Hourly  Diif.    =  2  .18  T=  11  22 8^2 

which  agrees  within  4*  with  the  true  value.  Taking  it  as  a  first 
approximation,  and  proceeding  as  in  Art.  55,  a  second  approxima- 
tion gives  T=  11*  22m  12M9. 

THE  EPHEMERIS,  OR  NAUTICAL  ALMANAC. 

57.  We  have  already  had  occasion  to  refer  to  the  Ephemeris ; 
but  we  propose  here  to  treat  more  particularly  of  its  arrange- 
ment and  use. 

The  Astronomical  Ephemeris  expresses  in  numbers  the  actual 
state  of  the  celestial  sphere  at  given  instants  of  time ;  that  is, 
it  gives  for  such  instants  the  numerical  values  of  the  co-ordi- 
nates of  the  principal  celestial  bodies,  referred  to  circles  whose 
positions  are  independent  of  the  diurnal  motion  of  the  earth, 
as  declination  and  right  ascension,  latitude  and  longitude; 
together  with  the  elements  of  position  of  the  circles  of  re- 
ference themselves.  It  also  gives  the  effects  of  changes  of  posi- 
tion of  the  observer  upon  the  co-ordinates,  or,  rather,  numbers 
from  which  such  changes  can  be  readily  computed  (namely, 
the  parallax,  which  will  be  fully  considered  hereafter),  the  ap- 
parent angular  magnitude  of  the  sun,  moon,  and  planets,  and, 
in  general,  all  those  phenomena  which  depend  on  the  time;  that 
is,  which  may  be  regarded  simply  as  functions  of  the  time. 

The  American  Ephemeris  is  composed  of  two  parts,  the  first 
computed  for  the  meridian  of  Greenwich,  in  conformity  with  the 
British  Nautieal  Almanac,  especially  for  the  use  of  navigators ; 
the  second  computed  for  the  meridian  of  Washington  for  the 
convenience  of  American  astronomers.  The  French  Ephemeris, 
La  Connaissance  des  Temps,  is  computed  for  the  meridian  of  Paris; 
the  German,  Berliner  Astronomisches  Jahrluch,  for  the  meridian 
of  Berlin.  All  these  works  are  published  annually  several  years 
in  advance. 

58.  In  what  follows,  we  assume  the  Ephemeris  to  be  computed 
for  the  Greenwich  meridian,  and,  consequently,  that  it  contains 
the  right  ascensions,  declinations,  equation  of  time,  £c.  for  given 
equidistant  instants  of  Greenwich  time. 


EPIIEMERIS.  69 

Before  we  can  find  from  it  the  values  of  any  of  these  quanti- 
ties for  a  given  local  time,  we  must  find  the  corresponding  Green- 
wich time  (Arts.  45,  46).  When  this  time  is  exactly  one  of  the 
instants  for  which  the  required  quantity  is  put  down  in  the  Ephe- 
meris,  nothing  more  is  necessary  than  to  transcrihe  the  quantity 
as  there  put  down.  But  when,  as  is  mostly  the  case,  the  time 
falls  between  two  of  the  times  in  the  Ephemeris,  we  must  obtain 
the  required  quantity  by  interpolation.  To  facilitate  this  inter- 
polation, the  Ephemeris  contains  the  rate  of  change,  or  difference 
of  each  of  the  quantities  in  some  unit  of  time. 

To  use  the  difference  columns  with  advantage,  the  Greenwich 
time  should  be  expressed  in  that  unit  of  time  for  which  the 
difference  is  given :  thus,  when  the  difference  is  for  one  hour, 
our  time  must  be  expressed  in  hours  and  decimal  parts  of  an 
hour ;  when  the  difference  is  for  one  minute,  the  time  should  be 
expressed  in  minutes  and  decimal  parts,  &c. 

59.  Simple  interpolation. — In  the  greater  number  of  cases  in 
practice,  it  is  sufficiently  exact  to  obtain  the  required  quantities 
by  simple  interpolation;  that  is,  by  assuming  that  the  differences 
of  the  quantities  are  proportional  to  the  differences  of  the  times, 
which  is  equivalent  to  assuming  that  the  differences  given  in  the 
Ephemeris  are  constant.  This,  however,  is  never  the  case;  but 
the  error  arising  from  the  assumption  will  be  smaller  the  less 
the  interval  between  the  times  in  the  Ephemeris ;  hence,  those 
quantities  which  vary  most  irregularly,  as  the  moon's  right 
ascension  and  declination,  are  given  for  every  hour  of  Green- 
wich time ;  others,  as  the  moon's  parallax  and  semidiameter,  for 
every  twelfth  hour,  or  for  noon  and  midnight ;  others,  as  the 
sun's  right  ascension,  &c.,  for  each  noon ;  others,  as  the  right 
ascensions  and  declinations  of  the  fixed  stars,  for  every  tenth  day 
of  the  year.  Thus,  for  example,  the  greatest  errors  in  the  right 
ascensions  and  declinations  found  from  the  American  Ephe- 
meris by  simple  interpolation  are  nearly  as  follows : — 


Sun 

Error  in  R.  A. 

OM 

Error  in  Decl. 

3".5 

Moon 

0.1 

1  .5 

Jupiter 
Mars 

0.1 
0.4 

0  .6 
2  .4 

Venus 

0.2 

5  .4 

70 


EPHEMERIS. 


To  illustrate  simple  interpolation  when  the  Greenwich  time  is 
given,  we  add  the  following 

EXAMPLES. 

For  the  Greenwich  mean  time  1856  March  30,  17*  II"1  12% 
find  the  following  quantities  from  the  American  Ephemeris : 
the  Equation  of  time,  the  Right  Ascension,  Declination,  Hori- 
zontal Parallax,  and  Semidiameter  of  the  Sun,  the  Moon,  and 
Jupiter. 


1.   The  Equation  of  time.— The  Gr.  T.  =  March  30,  17*  llw.2  =  March 
30,  17M87. 

H.D.  =  —  0-.763 
17.19 


(Page  11)^  at  moan  noon  =     -f-  4m  27'.11 
Corr.  for  17M9  =         —  13  .11 


E=    +  4     14.00 


—   13.11 


NOTE. — Observe  to  mark  E  always  with  the  sign  which  denotes  how  it  is  to  be 
applied  to  apparent  time.  If  increasing,  the  H.  D.  (hourly  difference)  should  have 
the  same  sign  as  E ;  otherwise,  the  contrary  sign. 


2.  Sun's  E.  A.  and  Dec. 

(P.  II.)     «atO*  =      0*  36">40'.78 


Corr.  for  17MS7  == 


2    36.29 


a=       0  39    17.07 

3  at  0*  =  -f  3°  57'  21".9 

Corr.  for  17M87  =  ±  __  16  39  .4 

9  =  -f  4    14     1  .3 


3.  Moon's  R.  A.  and  Dec. 
a  at  17*  = 
Corr.  for  11*.  2  = 


20* 

-f 


m  9».80 
27  .97 


±±i      20  18  37  .77 


d  at  17*  = 
Corr.  for  11TO.2  — 


25°  3'  10".9 
-f    1  32  .7 


=  —  25    1  38  .2 


H.  D.  -f  9-.094 
17.187 


H.D.  -f  58".15 
17.187 

999.4 


Diff. 


2'.4975 
11.2 

27.97 


Diff.  lm  -j-  8".275 

11.2 

92.68 


4.  Moon's  Hor.  Par.  (=  r)  and  Semid.  (=  S). 

TT  at  12*  =  58'  44".l  H.  D.  -f-     2".17 

Corr.  for  5*.2  ==  -f   11  .3  5.2 

TT  ==  58  55  .4  11.28 


EPHEMERIS.  71 

S  at  12*  =  16'  2".0  Diff.  in  12*  ==  +  7".l 

Corr.  for  5*.2  =  -f  3  .1 
£  =  16  5  .1 

5.  Jupiter  s  E.  A.  and  Dec, 

a  at  0*  =  23*  29"  49'.95  H.  D.  +  2M75 

Corr.  for  17M87  =5         +    37.38  17.187 

«  =  23  30    27.33  37.38 

d  at  0*  =  —  4°  22'  45".6  H.  D.  -f  13".74 

Corr.  for  17M87  =       -f     3  56  .1  17.187 

S  =  —  4     18  49  .5  236.1 

6.  Jupiter's  Hor.  Par.  and  Semid. — At  the  bottom  of  page  231,  we 
find  for  the  nearest  date  March  31,  without  interpolation : 

TT  ==  1".5  S  =  15".7 

NOTE. — It,  may  be  observed  that  we  mark  hourly  differences  of  declination  plus, 
when  the  body  is  moving  northward,  and  minus  when  it  is  moving  southward. 

In  the  above  we  have  carried  the  computation  to  the  utmost 
degree  of  precision  ever  necessary  in  simple  interpolation. 

00.  To  find  the  right  ascension  and  declination  of  the  sun  at  the  time 
of  its  transit  over  a  given  meridian,  ami  also  the  equation  of  time  at 
the  same  instant. 

When  the  sun  is  on  a  meridian  in  west  longitude,  the  Green- 
wich apparent  time  is  precisely  equal  to  the  longitude,  that  is, 
the  Gr.  App.  T.  is  after  the  noon  of  the  same  date  with  the  local 
date,  by  a  number  of  hours  equal  to  the  longitude.  When  the 
sun  is  on  a  meridian  in  east  longitude,  the  Gr.  App.  T.  is  before 
the  noon  of  the  same  date  as  the  local  date,  by  a  number  of 
hours  equal  to  the  longitude.  Hence,  to  obtain  the  sun's  right 
ascension  and  declination  and  the  equation  of  time  for  apparent 
noon  at  any  meridian,  take  these  quantities  from  the  Ephemeris 
(page  I  of  the  month)  for  Greenwich  Apparent  Noon  of  the 
same  date  as  the  local  date,  and  apply  a  correction  equal  to  the 
hourly  difference  multiplied  by  the  number  of  hours  in  the  lon- 
gitude, observing  to  add  or  subtract  this  correction,  according  as 
the  numbers  in  the  Ephemeris  may  indicate,  for  a  time  before  or 
after  noon. 


72 


EPHEMERIS. 


EXAMPLE  1.— Longitude  167°  31'  W.  1856  March  20,  App. 
Noon,  find  O's  R.  A.,  O's  Dec.,  and  Eq.  of  T. 

Longitude  =  -f-  11*  10'*  4'  =  -f  11M7 

a  at  App.  0*=   0*  0'*  20'.94      H.  D.  -f  9-.098 

Corr.  for  -f  11M7  =+   1    41.62  -f    11.17 

a=    0   2      2.56  -f  101.62 

d  at  App.  0»=  +  0°  2'  16".5       H.  D.  -f  59".21 

Corr.  for  +  11M7  =  -f       11     1  .4  -f-    11.17 

d  =  -f  0  13  17  .9  -f   661.4 

E  at  App.  0»  =  -f   7m  318.57      II.  D.  —  (K759 
Corr.  for  -f-  11M7  =  _  8.48 

E  =  +     7  2~iU)9~ 

EXAMPLE  2;— Longitude  167°  31'  E.  1856  March  20,  App. 
Noon,  find  O's  R.A.,  O's  Dec.,  and  Eq.  of  T. 

Longitude  =  —  11*  10m  4'=  —  11M7 

«  at  App.  0*  ==   0*  0-  20«.94        II.  D.  -f  9'.098 

Corr.  for  -  11M7  =_-  -  1   41 .62  11.17 

«  =  23  58   39.32  —101.62 

d  at  App.  0*  =  +   0°  2'  16".5        II.  D.  -f  59".21 

Corr.  for  —  11M7  =  11     1  .4  11.17 

*  d  =  —  0     8  44  .9  —    661.4 

E  at  App.  0A  =      -f  7ni  31-.57        H.  D.  —  0'.759 
Corr.  for  —  11*.17  =          +      8 .48  11.17 

E  =      +"7    40.05  +      8.48 

61.  To  find  the  mean  local  time  of  the  moon's  or  a  planet's  transit 
over  a  given  meridian. 

This  is  the  same  as  the  problem  of  Art.  55,  in  the  special  case 
where  the  hour  angle  of  the  moon  or  planet  at  the  given  meri- 
dian is  O7'.  We  can,  however,  obtain  the  required  time  directly 
from  the  Ephemeris,  with  sufficient  accuracy  for  many  purposes, 


*  In  this  example  the  sun  crosses  the  equator  between  the  times  of  its  transits 
•  over  the  local  and  the  Greenwich  meridians.  The  case  must  be  noted,  as  it  is  a  fre- 
quent occasion  of  error  among  navigators.  The  same  case  can  occur  on  September 
.22  or  23. 


EPIIEMERIS.  73 

by  simple  interpolation.  On  page  IV  of  the  month  (Am.  Ephem. 
and  British  Naut.  Aim.}  we  find  the  mean  time  of  transit  of  the 
moon  over  the  Greenwich  meridian  on  each  day.  This  mean 
time  is  nothing  more  than  the  hour  angle  of  the  mean  sun  at 
the  instant,  or  the  difference  of  the  right  ascensions  of  the  moon 
and  the  mean  sun ;  and  if  this  difference  did  not  change,  the 
mean  local  time  of  moon's  transit  would  he  the  same  for  all 
meridians ;  hut  as  the  moon's  right  ascension  increases  more 
rapidly  than  the  sun's,  the  moon  is  apparently  retarded  from 
transit  to  transit.  The  difference  between  two  successive  times 
of  transit  given  in  the  Ephemeris  is  the  retardation  of  the  moon 
in  passing  over  247t  of  longitude,  and  the  hourly  difference  given 
is  the  retardation  in  passing  from  the  Greemvich  meridian  to 
the  meridian  1A  from  that  of  Greenwich.  Hence,  to  find  the 
local  time  of  the  moon's  transit  on  a  given  day,  take  the  time  of 
meridian  passage  from  the  Ephemeris  for  the  same  date  (astro- 
nomical account)  and  apply  a  correction  equal  to  the  hourly 
difference  multiplied  by  the  longitude  in  hours;  adding  the 
correction  when  the  longitude  is  west,  subtracting  it  when  east. 
The  same  method  applies  to  planets  whose  mean  times  of  transit 
are  given  in  the  Ephemeris  as  in  the  case  of  the  moon. 

EXAMPLE.— Longitude  130°  25'  E.  1856  March  22 ;  required 
local  time  of  moon's  transit. 

Gr.  Merid.  Passage  March  22,  13*.  2m.7      II.  D.  -f  1-.59 
Corr.  for  Long.  —  8*.7      =          -  13.8  8.7 

Local  M.  T.  of  transit      =        12  48.9  —    13.8 

62.  To  find  the  moon's  or  a  planet's  right  ascension,  declination, 
$c.,  at  the  time  of  transit  over  a  given  meridian. 

Find  the  local  time  of  transit  by  the  preceding  article,  deduce 
the  Greenwich  time,  and  take  out  the  required  quantities  from 
the  Ephemeris  for  this  time.  This  is  the  usual  nautical  method, 
and  is  accurate  enough  even  for  the  moon,  as  meridian  observa- 
tions of  the  moon  at  sea  are  not  susceptible  of  great  precision. 
For  greater  precision,  find  the  local  time  by  Art.  55  for  /  =  0/4, 
and  thence  the  Greenwich  time.  See  also  Moon  Culminations, 
Chapter  VII. 

63.  INTERPOLATION  BY  SECOND  DIFFERENCES. — The  differences 
between  the  successive  values  of  the  quantities  given  in  the 


74  EPHEMERIS. 

Ephemeris  as  functions  of  the  time,  are  called  the  first  differ- 
dices;  the  differences  between  these  successive  differences  are 
called  the  second  differences ;  the  differences  of  the  second  differ- 
ences are  called  the  third  differences,  &c.  In  simple  interpolation 
we  assume  the  function  to  vary  uniformly ;  that  is,  we  regard 
the  iirst  difference  as  constant,  neglecting  the  second  difference, 
which  is,  consequently,  assumed  to  be  zero.  In  interpolation 
by  second  differences  we  take  into  account  the  variation  in  the 
first  difference,  but  we  assume  its  variations  to  be  constant; 
that  is,  we  assume  the  second  differences  to  be  constant  and  the 
third  differences  to  be  zero. 

When  the  American  Ephemeris  is  employed,  we  can  take  the 
second  differences  into  account  in  a  very  simple  manner.  In 
this  work,  the  difference  given  for  a  unit  of  time  is  always  the 
difference  belonging  to  the  instant  of  Greenwich  time  against 
which  it  stands,  and  it  expresses,  therefore,  the  rate  at  which 
the  function  is  changing  at  that  instant.  This  difference,  which 
we  may  here  call  the  first  difference,  varies  with  the  Greenwich 
time,  and  (the  second  difference  being  constant)  it  varies  uni- 
formly, so  that  its  value  for  any  intermediate  time  may  be  found 
by  simple  interpolation,  using  the  second  differences  as  first  dif- 
ferences. Now,  in  computing  a  correction  for  a  given  interval 
of  Greenwich  time,  we  should  employ  the  mean,  or  average 
value,  of  the  first  difference  for  the  interval,  and  this  mean 
value,  when  we  regard  the  second  differences  as  constant,  is 
that  which  belongs  to  the  middle  of  the  interval.  Hence,  to 
take  into  account  the  second  differences,  we  have  only  to  observe 
the  very  simple  rule — employ  that  (interpolated)  value  of  the  first 
difference  which  corresponds  to  the  middle  of  the  interval  for  'which  the 
correction  is  to  be  computed. 

EXAMPLE.— For  the  Greenwich  time  1856  March  2, 12*  29W  36', 
find  the  moon's  declination. 

March  2,  12*  (d)  =  —  27°  10'41".8  Diff.  I™  =  4.  4".814     2d  Diff.  =  -f  0".189 

Corr.  for  29'".  6  -f-     2  23  .9         Corr.  for  2<ldiff.  +  .047  0.25 

6=  —  27     8  17  .9  -f- 4.861  -f-  0.047 

29.6 
+  1:43.89 

Here  the  "cliff,  for  l"ff  increases  0".189  in  1* ;  the  half  of  the 
interval  for  which  the  correction  is  to  be  computed  is  14wl  48*  = 


EPHEMERIS.  75 

0\25;  we  therefore  find  the  value  of  the  first  difference  at  12'' 
14"1  48',  by  adding  to  its  value  taken  for  12''  the  quantity  0".189 
X  0.25,  and  then  proceed  as  in  simple  interpolation.  This  exam- 
ple suffices  to  illustrate  the  method  in  all  cases  where  the  first 
difference  is  given  in  the  Ephemeris  for  the  time  against  which 
it  stands.  In  using  the  British  Nautical  Almanac  and  other 
works  of  the  same  kind,  interpolation  by  second  differences 
may  be  performed  by  the  general  interpolation  formula  here- 
after given. 

64.  To  find  the  Greenwich  time  corresponding  to  a  given  right  ascen- 
sion of  the  moon  on  a  given  day. 

Let  T'  =  the   Greenwich  time  corresponding  to  the  given  right 

ascension  a', 
T  =  the  Greenwich  hour  preceding  T'  and  corresponding  to 

the  right  ascension  a, 
Aa  =  the  diff.  of  K.  A.  in  1-  at  the  time  T, 

then  we  have,  approximately, 

rjif  ?TI 

Aa 

To  correct  for  second  differences,  we  have  now  only  to  find 

A()a  =  diff.  of  R.A.  in  lm  for  the  middle  instant 
of  the  interval  T'—T, 

and  then  we  have,  accurately, 

/ 

rnt  rn  tt  a 

~J^T 
These  formulae  snve  T'  —  T  in  minutes  of  time. 


65.  To  find  the  distance  of  the  moon  from  a  given  object  at  a  given 
Greenwich  time. 

In  the  American  Ephemeris  and  the  British  Nautical  Alma- 
nac, the  "  lunar  distances"  are  given  at  every  3d  hour  of  Green- 
wich time,  together  with  the  proportional  logarithms  of  the  differ- 
ences between  the  successive  distances. 

The  proportional  logarithm  of  an  angle  expressed  in  hours, 
£c.  is  the  logarithm  of  the  quotient  of  3*  divided  by  the  angle  ; 
that  of  an  angle  expressed  in  degrees,  &c.  is  the  logarithm  of 
the  quotient  of  3°  divided  by  the  angle.  Thus,  if  A  is  the  angle, 
in  hours, 


76  EPHEMERIS. 

3* 
.r .  ±j.  A  =  log  —  =  log  o*    —  log  A 

A 
or,  if  A  is  in  degrees, 

3° 
P.  L.  A  =  log  —  =  log  3°  —  log  A 

The  angle  is  always  supposed  to  be  reduced  to  seconds ;  so  that, 
whether  A  is  in  seconds  of  time  or  of  arc,  we  have 

P.  L.  A  =  log  10800  —  log  A 

Tables  of  such  logarithms  are  given  in  works  on  Navigation. 

If  now  we  wish  to  interpolate  a  value  of  a  lunar  distance  for  a 
time  T -\- 1  which  falls  between  the  two  times  of  the  Ephemeris 
Tand  T-\-  37',  we  are  to  compute  the  correction  for  the  interval  t 
and  apply  it  to  the  distance  given  for  the  time  T\  and  if  we  put 

A  =the  difference  of  the  distances  in  the  Ephemeris, 
A'  =  the  difference  in  the  interval  f, 

we  shall  have,  by  simple  interpolation, 

or,  by  logarithms, 

log  A'  =  log  t  -f  log  A  —  log  3* 

or,  supposing  J,  J',  and  t  all  reduced  to  seconds, 

log  A'  =  log  t  —  P.  L.  A  (62) 

Subtracting  both  members  of  this  from  log  10800,  we  have 

P.  L.  J'  =  P.L.  t  +  P.  L.  A  (63) 

which  is  computed  by  the  tables  above  mentioned.  By  (62), 
however,  only  the  common  logarithmic  table  is  required. 

But  the  first  differences  of  the  lunar  distance  cannot  be  assumed 
as  constant  when  the  intervals  of  time  are  as  great  as  3*.  If 

we  put 

P.  L.  A  =  Q 

we  observe  that  Q  is  variable,  and  the  value  given  in  the  Ephe- 
meris is  to  be  regarded  as  its  value  at  the  middle  instant  of  the 
interval  to  which  it  belongs.  If  then 

Q'  =the  value  of  Q  for  the  middle  of  the  interval  f, 
A§  =  the  increase  of  Q  in  3*  (found  from  the  successive  values 
in  the  Ephemeris), 


EPHEMERIS.  77 

we  have 

(64) 


in  which  t  is  in  hours  and  decimal  parts.     We  find  then,  with 
regard  to  second  differences, 

log  JT^  log  *_.£'' 

EXAMPLE.  —  Find  the  distance  d  of  the  moon's  centre  from  the 
star  Fomalhaut  at  the  Greenwich  time  1856  March  30,  13*  20'* 
24*. 

Here  T=  12*,*  =  1»20»  24s  =  1».34;  1>t5  ~  **  =  0.28;  and  from  the 
Ephemeris  : 

March  30,  12*  (d)        36°  17'  53"        §,       .2993     A§,  +  .0041 

J'       -0  40  28  -.0011  .28 

At  13»  20-  24"  d  =  35  37  25    Q',   .2982      -f  .0011 

log*,  3.6834 

log  J',  3.3852 


66.  Jb  ^??.d?  the  Greenwich  time  corresponding  to  a  given  lunar  dis- 
tance on  a  given  day. 

We  find  in  the  Ephemeris  for  the  given  day  the  two  distances 
between  which  the  given  one  falls;  and  if  A*  =  difference  be- 
tween the  first  of  these  and  the  given  one,  J  =  difference  of  the 
distances  in  the  Ephemeris,  we  find  the  interval  £,  to  be  added  to 
the  preceding  Greenwich  time,  by  simple  interpolation,  from  the 
formula 

*  =  3»x- 

J 

or 

log  t  =  log  J'  +  P.  L.  A  =log  A'  -J-  Q  (65) 

and,  with  regard  to  second  differences,  the  true  interval,  t'  ',  by 
the  formula 

logf  =  logJ'+Q'  (66) 

where  Q'  has  the  value  given  in  the  preceding  article. 

But  to  find  Q'  by  (64)  we  must  first  find  an  approximate  value 
of  t.  To  avoid  this  doublo  computation,  it  is  usual  to  find  t  by 
(65),  and  to  give  a  correction  to  reduce  it  to  t'  in  a  small  table 
which  is  computed  as  follows.  We  have  from  (64),  (65),  and  (66) 


78 


EPIIEMERIS. 


By  the  theory  of  logarithms,  we  have,  M  being  the  modulus 
of  the  common  system, 

log  x  =  M[(x  —  1)  —  l(x  —  I)2  +  &c.] 
so  that 

/'  r/'  _  /  1    /  /'  _  /  \2  T 

log  f  -  log  t  =  log  L  =*  PJ-*  -  jjj  -p  )  +  &c'] 

or,  neglecting  the  square  and  higher  powers  of  the  small  fraction 
t'  —  t 


This,  substituted  above,  gives 


by  which  a  table  is  readily  computed  giving  the  value  of  t'  —  t 
[or  the  correction  of  t  found  by  (65)],  with  the  arguments  A§  and  t. 
In  this  formula  t  and  V  —  t  are  supposed  to  be  expressed  in  hours; 
and  to  obtain  t'  —  t  in  seconds  we  must  multiply  the  second 
member  by  3600  ;  this  will  be  effected  if  we  multiply  each  of  the 
factors  t  and  3A  —  t  by  60,  that  is,  reduce  them  each  to  minutes, 
so  that  if  we  substitute  the  value  of  M=  .434294  the  formula 
becomes 

)  (67) 


2.60576 
in  which  t  is  expressed  in  minutes,  and  t'  —  t  in  seconds. 

EXAMPLE.  —  1856  March  30,  the  distance  of  the  moon  and 
Fomalhaut  is  35°  37'  25"  ;  what  is  the  Greenwich  time  ? 

March  30,  12*   O"1   0*  (d)  =  36°  17'  53"         Q=  .2993  A§=  +  41 

f=   1  20  36     d  =35   37  25  log  J'  =  3.3852 
Ap.  Gr.  time  =13  20  36     J'  4028    log  t   =  3.6845 

By(67)V—  f  =  _      -12 
True  Gr.  time  —  13  20  24 

*  Or  from  the  "  Table  showing  the  correction  required  on  account  of  the  second 
differences  of  the  moon's  motion  in  finding  the  Greenwich  time  corresponding  to  a 
corrected  lunar  distance,"  which  is  given  in  the  American  Ephemeris,  and  is  also 
included  in  the  Tables  for  Correcting  Lunar  Distances  given  in  Vol.  II.  of  this  work. 


INTERPOLATION    IN    GENERAL. 


79 


INTERPOLATION    BY    DIFFERENCES    OF   ANY    ORDER. 

67.  When  the  exact  value  of  any  quantity  is  required  from  the 
Ephemeris,  recourse  must  be  had  to  the  general  interpolation 
formulae  which  are  demonstrated  in  analytical  works.  These 
enable  us  to  determine  intermediate  values  of  a  function  from 
tabulated  values  corresponding  to  equidistant  values  of  the 
variable  on  which  they  depend.  In  the  Ephemeris  the  data  are 
in  most  cases  to  be  regarded  as  functions  of  the  time  considered 
as  the  variable  or  argument. 

Let  J7,  T-{-  w,  T-\-  2w,  T-\-  3?/;,  &c.,  express  equidistant  values 
of  the  variable ;  F,  Ff,  Fn ',  F"',  &c.,  corresponding  values  of 
the  given  function ;  and  let  the  differences  of  the  first,  second, 
and  following  orders  be  formed,  as  expressed  in  the  following 
table :— 


Argument. 

Function. 

1st  DifF. 

2d  DifF. 

3d  DifF. 

4th  DifF. 

5th  DifF. 

T 

F 

a 

T+    w 

F' 

b 

a' 

C 

T  -f-  2w 

F" 

b' 

d 

a" 

c' 

e 

T  -f  870 

F'" 

b" 

d' 

a'" 

c" 

ef 

T  _L  4.w 

P\v 

b'" 

d" 

r 

v 

a* 

iv 

c'" 

+  OJ0 

a" 

T-i-Gw 

^vi 

f 


The  differences  are  to  be  found  by  subtracting  downwards,  that 
is.  each  number  is  subtracted  from  the  number  below  it,  and  the 
proper  algebraic  sign  must  be  prefixed.  The  differences  of  any 
order  are  formed  from  those  of  the  preceding  order  in  the  same 
manner  as  the  first  differences  are  formed  from  the  given  func- 
tions. The  even  differences  (2d,  4th,  &c.)  fall  in  the  same  lines 
with  the  argument  and  function  ;  the  odd  differences  (1st,  3d,  &c.) 
between  the  lines. 

Now,  denoting  the  value  of  the  function  corresponding  to  a 
value  of  the  argument  T-\-  mo  by  F(H\  we  have,  from  algebra, 


c.     (68) 


1.2  1.2.3  1.2.3.4 

in  which  the  coefficients  are  those  of  the  nth  power  of  a  binomial. 


80 


INTERPOLATION    IN    GENEKAL. 


In  this  formula  the  interpolation  sets  out  from  the  first  of  the 
given  functions,  and  the  differences  used  are  the  first  of  their 
respective  orders.  If  n  be  taken  successively  equal  to  0,  1,  2,  3, 
&c.,  we  shall  obtain  the  functions  F,  F',  F",  F'",  &c.,  and  in- 
termediate values  are  found  by  using  fractional  values  of  n.  We 
usually  apply  the  formula  only  to  interpolating  between  the 
function  from  which  we  set  out  and  the  next  following  one,  in 
which  case  n  is  less  than  unity.  To  find  the  proper  value  of  n 
in  each  case,  let  T-\- 1  denote  the  value  of  the  argument  for  which 
we  wish  to  interpolate  a  value  of  the  function :  then 

nw  =  t  n  =  - 

w 

that  is,  n  is  the  value  of  t  reduced  to  a  fraction  of  the  interval  w. 

EXAMPLE. — Suppose    the    moon's   right   ascension   had  been 
given  in  the  Ephemeris  for  every  twelfth  hour  as  follows : 

5th  Diff. 


—  C'.GG 


Required  the  moon's  right  ascension  for  March  5,  6*. 

Here  T=  March  5,  O7',  t  =  6\  w  =  12*,  n  =  —       1;   and  if  we 

i  —       L 

denote  the  coefficients  of  a,  6,  c,  d,  e  in  (68)  by  A,  B,  C,  D,  E, 

we  have 

.F  =  21*  58™  28«.39 

a  =  +  28-  47*.04,     A  =  n  \,     Aa 


P's  R.  A. 

1st.  Diff. 

2d  Diff. 

3d  Diff. 

4th  Diff. 

1856  March  5,     0* 

21*  58m  28«.39 

+  28m  47'.  04 

k 

"       5,  12 

22   27    15.43 

—  36*.  97 

28    10.07 

+  4'.  79 

"      6,     0 

22   55    25.50 

32.18 

+  K74 

27     37  .89 

6.53 

"      6,  12 

23   23       3.  39 

25.65 

1.08 

27    12.24 

7.61 

"      7,     0 

23   50    15.  G3 

18.04 

26    54.20 

"      7,  12 

0   17      9.83 

36.97,     5= 


b=  — 
c=+. 

d  =  +  1.74,    D  = 

e=—  0.66,     E  = 


-f    14    23.52 
4-  4.62 


4.79,     C=B.^—^  =  + 


Cc  =  + 


0.30 
0.07 
0.02 


>'s  E.  A.  1856  March  5,  6' 


=  22   12    56  .74 


INTERPOLATION. 


81 


which  agrees  precisely  with  the  value  given  in  the  American 
Ephemeris. 

68.  The  formula  (68)  may  also  he  written  as  follows : 


Thus,  in  the  preceding  example,  we  should  have 
n— 4 


5 
n— 3 

4 
n—2 

3 
n  — 1 


-  T7u  X  -  0-.66  = 

_  5  (_[_    i«j4  +  Q-.46)       = 

_  £  (-|-    4-.79  _  K38)       =±= 

_  i  (_  36«.97  rrr  I'.Tl)       = 
i  (+  28-47«.04-f  9-.67)  = 


+  0-.46 

-  1.38 

—  1.71 

+  9.67 
4-  28'.35 


and  adding  this  last  quantity,  14"*  28*.35,  to  2lh  58W  28*.39,  we 
obtain  the  same  value  as  before,  or  22*  12"1  56*.74. 

69.  A  more  convenient  formula,  for  most  purposes,  may  be 
deduced  from  (68),  if  we  use  not  only  values  of  the  functions 
fallowing  that  from  which  we  set  out,  but  also  preceding  values; 
that  is,  also  values  corresponding  to  the  arguments  T  —  w, 
T  —  2m,  &c.  We  then  form  a  table  according  to  the  following 
schedule : 


rgument. 

Function. 

1st  Diff. 

2dDiff. 

3d  Diff. 

4th  Diff. 

5th  Diff. 

6th  Diff. 

T  —  3w 

#* 

T—  2w 

*« 

'" 

Q 

T—   w 

^ 

a" 

Q 

CH 

dj 

ai 

c. 

*, 

T 

JP 

b 

d 

f 

a' 

cf 

e' 

T+   w 

J« 

V 

d' 

a" 

c" 

T+2w 

^rl 

b" 

a'" 

T+3^ 

fW* 

VOL.  I.—  (f 

82  INTERPOLATION. 

According  to  the  formula  (68),  if  we  set  'out  from  the  function 
F,  we  employ  the  differences  denoted  in  this  table  by  «',  &',  c", 
&c.,  and  hence  for  the  argument  T  -\-  nw  we  find  the  value  of 
F^  by  the  formula 


f 


1.2  1  .  2  .  3  1.2.3.4 

But  we  have 

V  =  b  +c' 

c"  =  <f  -f  d'  =  c'  -f  d  -f  e' 

d"  =  d'  -f  e"  =  d  -f  e'  +  e'  +  /'  ==  d  +  2e' 

&c.         &c.  i~)  i 

in  which  £',  e",  &c.  are  expressed  in  terms  of  the  differences 
that  lie  on  each  side  of  a  horizontal  line  drawn  in  the  table 
immediately  under  the  function  from  which  we  set  out.  These 
values  substituted  in  the  formula  give 


1.2  1.2.3 

--2) 


1.2.3.4 

in  which  the  law  of  the  coefficients  is  that  one  new  factor  is 
introduced  into  the  numerator  alternately  after  and  before  the 
other  factors,  observing  always  that  the  factors  decrease  by  unity 
from  left  to  right.  The  new  factor  in  the  denominator,  as  in  the 
original  formula  (68),  denotes  the  order  of  difference. 

The  interpolation  by  this  formula  is  rendered  somewhat  more 
accurate  by  using,  instead  of  the  last  difference,  the  mean  of  the 
two  values  that  lie  nearest  the  horizontal  line  drawn  under  the 
middle  function:  thus,  if  we  stop,  at  the  fourth  difference,  we 
use  a  mean  between  c?:and-d'  instead  of  d.  We  thus  take  into 
account  a  part  df  the  term  involving  the  fifth  'difference. 

I      ' 

EXAMPLE.—  Find  the  moon's  right  ascension  for  1856  March  5, 
6*,  employing  the  values  given  in  the  Ephemeris  for  every 
twelfth  hour.  This  is  the  same  as  tlje  example  under  Art.  67, 
where  it  is  worked  by  the  primitive  formula  (68).  But  here  we 
take  from  the  Ephemeris  three  values'  preceding  that  for  March  5, 
O7',  and  three  values  following  it,  and  form  our  table  as  follows: 


INTERPOLATION. 


83 


1856  March  3,  12* 
"       4,    0 
"       4,  12 
"       5,    0 

D's  R.  A. 

1st  Diff. 

2d  Diff. 

3d  Diff. 

4th  Diff. 

5th  Diff. 

20*  28"*  17'.  88 
20   58    57.08 
21    29      2.01 
21    58    28.39 

-f  30»»39».20 
30  4.93 
29  26.38 

—  34».27 

38.55 
39.34 

—  4'.  28 
—  0.79 

-f  3'.  49 
3.16 

—  0».33 

"       5,  12 
6,    0 
"       6,  12 

22   27    15.43 
22   55    25.50 
23   23      3.39 

28  47.04 
28  10.07 
27  37.89 

36.97 
32.18 

4-2.37 
+  4.79 

2.42 

—  0.74 

Drawing  a  horizontal  line  under  the  function  from  which  we 
set  out,  the  differences  required  in  the  formula  (69)  stand  next 
to  this  line,  alternately  below  and  above  it. 


«'  =  + 

h  • 

28-  47-.04, 
39  .34, 

2.37, 
3.16, 

0.74, 
1 
If  in  (69) 

A=         n         =         i, 

joL^   !Lnlt= 

Aa'  =  +    14 
56—4- 

23.52 
4.92 

0.15 
0.07 
0.01 

<>  =  + 

rl  -  _U 

2 

c-j?   n+1- 

JJU                          l 

Cc'  —  — 

7)/7           _j_ 

3 

D       C     n~~2         1      » 

a  —  rp 
f" 

^     '                    4                        "I          1^6' 

77       D     n  ~^~2         J      3 

Ee' 

69*. 
we  find 

-^         -V  •          r             -     »      :i£67 

>'s  R.  A.  1856  March  5,  6*  = 

we  substitute  the  values 
a'  =  at  +  b 

&c. 
(n  +  1)  n         (n  +  1) 

f\w  _  22   12 
(n)  (n  -  1) 

56  .74 

/"Tf^                    -N 

4-  ^ 

ld^         1.2                         1 

.2.3             C' 

-L  Xr.o 

1.2.3.4 

in  which  the  law  of  the  coefficients  is  that  one  new  factor  is 
introduced  into  the  numerator  alternately  before  and  after  the 
other  factors,  observing  still  that  the  factors  decrease  by  unity 
from  left  to  right.  The  differences  employed  are  those  which  lie 
on  each  side  of  the  horizontal  line  drawn  immediately  above 
the  function  from  which  we  set  out. 


84  INTERPOLATION. 

If  in  the  preceding  formulae  we  employ  a  negative  value  of 
n  less  than  unity,  we  shall  obtain  a  value  of  the  function  between 
F  and  Fn  and  in  that  case  (70)  is  more  convergent  than  (69).  In 
general,  if  we  set  out  from  that  function  which  is  nearest  to  the 
required  one,  we  shall  always  have  values  of  n  numerically  less 
than  J,  and  we  should  prefer  (69)  for  values  of  n  between  0  and 
-f  },  and  (70)  for  values  of  n  between  0  and  —  J. 

70.  If  we  take  the  mean  of  the  two  formula  (69)  and  (70), 
and  denote  the  means  of  the  odd  differences  that  lie  above  and 
below  the  horizontal  lines  of  the  table,  by  letters  without  ac- 
cents, that  is,  if  we  put 

'   a  =  *(«,  +  *')>    <?=*(*,  +  O&c- 
we  have 

p»=F+na+«  6+  (n+1)  W  fr-1*  c+  (n+1)  W  <*-D  d  +  &c.  (71) 
T2  2.3  2.3.4 

The  quantities  a,  c,  &e.  may  be  inserted  in  the  table,  and  will 
thus  complete  the  row  of  differences  standing  in  the  same  line 
with  the  function  from  which  we  set  out. 

The  law  of  the  coefficients  in  (71)  is  that  the  coefficient  of  any 
odd  difference  is  obtained  from  that  of  the  preceding  odd  dif- 
ference by  introducing  two  factors,  one  at  the  beginning  and 
the  other  at  the  end  of  the  line  of  factors,  observing  as  before 
that  these  factors  are  respectively  greater  and  less  by  unity  than 
those  next  to  which  they  are  placed;  and  the  coefficients  of  the 
even  differences  are  obtained  from  the  next  preceding  even 
differences  in  the  same  manner.  The  factors  in  the  denominator 
follow  the  same  law  as  in  the  other  formulae. 

EXAMPLE. — Find  the  moon's  right  ascension  for  1856  March  5, 
6*,  from  the  values  given  in  the  Ephemeris  for  noon  and  mid- 
night 

The  table  will  be  as  below: 


INTERPOLATION. 


85 


Mar.  3,  12* 
«     4,     0 
«     4,  12 

D's  R.  A. 

1st  Diff. 

2d  Diff. 

3d  Diff. 

4th  Diff. 

5th  Diff. 
—  0«.33 

20*28"*17'.88 
20  58  57  .08 
21  29     2  .01 

-f-30w29'.20 
30  4  .93 
29  26.38 

—  34*.27 
38.55 

—  4'.28 
-0.79 

+3».49 

«    C  o 

21  58  28.39 

[+29          6   .71] 

—  39.34 

[+0    .79] 

+  3.16 

[-0.54] 

"     5,  12 
"     6,     0 
"     6,  12 

22  27  15  .43 
22  55  25  .50 
23  23     3  .39 

28  47.04 
28  10.07 
27  37.89 

36.97 
32.18 

+  2.37 
+  4.79 

2.42 

-0.74 

Drawing  two  lines,  one  above  and  the  other  below  the  func- 
tion from  which  we  set  out,  and  then  tilling  the  blanks  by  the 
means  of  the  odd  differences  above  and  below  these  lines  (which 
means  are  here  inserted  in  brackets),  we  have  presented  in  the 
same  line  all  the  differences  required  in  the  formula  (71) ;  and 
we  then  have 

F=  21*  58*  28'.39 
a  =  -f  29-  6-.71,  A  =  n  ==        J,        Aa  =  -f-   14    33  .36 


6 
c 
d 
e 

i 

39 
0 
3 
0 

.34,  B  = 
7Q  f 

n 

¥ 

A. 
B. 

a. 

n2—  i 

t           /x. 

4 
0 
0 
0 

.92 
.05 
.02 
.01 

~r 
.    i 

.16,  D  = 

C  ±  IF 

6 
n2—  1 

T5J 
1          T),^ 

—  ~r 

12 
n2—  4 

jig?     -U(l 

.t)-±,  Jit  

20     ~    ' 

25g>     ^e 

=  22   12    56  .75 

agreeing  within  O'.Ol  with  the  value  found  in  the  preceding 
article.  HANSEN  has  given  a  table  for  facilitating  the  use  of  this 
formula.  (See  his  Tables  de  la  Lime). 

71.  Another  form,  considered  by  Bessel  as  more  accurate  than 
any  of  the  preceding,  is  found  by  employing  the  odd  differences 
that  fall  next  below  the  horizontal  line  drawn  below  the  function 
from  which  we  set  out,  and  the  means  of  the  even  differences 
that  fall  next  above  and  next  below  this  line.  Thus,  if  we  put 


bQ  = 


(b 


&c. 


86 


INTERPOLATION. 


and  combine  these  with  the  expressions 

we  deduce 

b  =  b0  —  £  cry          d  =  d0  —  J  e'j  &c. 

which  substituted  in  (69)  give 


F<«=F+na>+  .   , 

1.2  1.2.3 


1.2.3.4 


To  facilitate  the  application  of  this  formula,  draw  a  horizontal 
line  under  the  function  from  which  the  interpolation  sets  out, 
and  another  over  the  next  following  function;  these  lines  will 
embrace  the  odd  differences  a',  c',  &c.  If  we  then  insert  in  the 
blank  spaces  between  these  lines  the  means  of  the  even  differ- 
ences that  fall  above  and  below  them,  we  shall  have  presented 
in  a  row  all  the  differences  to  be  employed  in  the  formula. 

EXAMPLE.  —  Find  the  right  ascension  of  the  moon's  second 
limb  at  the  instant  of  its  transit  over  the  meridian  whose  longi- 
tude is  4;'  42'H  19*  west  from  Greenwich,  on  May  15,  1851. 

The  right  ascensions  of  the  moon's  bright  limb  at  the  instant 
of  its  upper  and  lower  transits  over  the  Greenwich  meridian,  are 
given  in  the  Ephemeris,  under  the  head  of  "Moon  Culminations." 
The  argument  in  this  case  is  the  longitude,  and  the  intervals  of 
the  argument  are  12\  The  value  for  any  meridian  is  therefore 
to  be  obtained  by  interpolation,  taking  for  n  the  quotient  obtained 
by  dividing  the  given  longitude  (in  hours)  by  12\ 

We  take  from  the  British  Nautical  Almanac  the  following 
values: 


R.  A.D's2dlimb. 

1st  Diff. 

2d  Diff. 

3d  Diff.T 

4th  Diff. 

5th  Diff. 

May  14,  U.  C. 
"  15,  L.  C. 
"  15,  U.  C. 

15*  12"'  39*.  04 
15  41  3.41 
16  9  39.89 

-|-  28"'  24'.  37 
28     36  .48 

-f  12M1 
-f    9.49 

—  2».62 

—  K58 

28     45  .97 

[+7.39] 

—  4.20 

f-1.42] 

-f-C'.33 

-  «  16,  L.  C. 
«  16,  U.  C. 
"  17,  L.  C. 

16  38  25.86 
17  7  17.12 
17  36  8.22 

28     51  .20 
28     51.10 

-f    5.29 
—    0.16 

—  5.45 

—  1.25 

INTERPOLATION. 


87 


For  interpolation  by  formula  (72)  we  draw  a  horizontal  lino 
below  tbe  function  from  which  we  set  out,  and  one  above  the 
next  following  function.  These  lines  enclose  the  odd  differences 
regularly  occurring  in  the  table.  Inserting  in  the  blanks  in  the 
columns  of  even  differences  the  means  of  the  numbers  above  and 
below,  all  the  differences  to  be  employed  in  the  formula  stand  in 
the  same  line,  namely  : 


=  +  1725'.97,  b0  =  +  7'.39,  c'  =  —  4*.20,  d0  = 


,  e'  =  +  0-.33 


As  n  is  here  not  a  simple  fraction,  the  computation  will  be 
most  conveniently  performed  by  logarithms,  as  follows  : 


4*  42-19'  =  16939' 
12*  =43200 


log  4.2288878 
.log  4.6354837 


log  A  =  log  n  =  9.5934041 

w  =   0.3921065 

9.59340 

9.5934 

9.5934 

9.5934 

1  =  —  0.60789 

n9.78383 

W9.7838 

n9.7838 

W9.7838 

$  =  —  0.10789 

7i9.0330 

7i9.0330 

2  =  —  1.6079 

wO.2063 

TiO.2063 

1  =  +  1.3921 

0.1437 

0.1437 

(A)   9.5934041 
(a')   3.2370332 

(i)  9.698-97 

(i)  9.2218 

(A)  8-6198 

(D)  8.3470 

(^0>0.1523 
n8.4993 

(T^)7.9208 

(5)n9.07620 
(60)  0.86864 

(C)  7.6320 

(c>0.6232 

(e;)  9.5185 
7i6.1995 

2.8304373 

7i9.94484 

wS.2552 

Aa'  =        llm  16'.764 

Bb0  =  —  0 .879 

Of   =—  0.018 

•DdQ  =  —  0 .032 

Eg  =  0.000 

Increase  of  E.  A.  11    15  .835 

E.  A.  Greenwich  Culm.  =  16*    9ro  39'.890 


E.  A.  on  given  meridian  =  16*  20"  55'.725 

The  use  of  BESSEL'S  formula  of  interpolation  is  facilitated  by  a 
table  in  which  the  values  of  the  coefficients  above  denoted  by 
A,  J5,  (7,  Dj  &c.,  and  also  their  logarithms,  are  given  with  the 
argument  n. 

•  72.  Interpolation  into  the  middle. — When  a  value  of  the  function 
is  sought  corresponding  to  a  value  of  the  argument  which  is  a 


88  INTERPOLATION. 

mean  between  two  values  for  which  the  function  is  given,  that 
is,  when  n  —  J,  we  have  by  (72),  since  n  —  J  =  0, 


-        +  7|g  rf0-T&5  /o  +  &c. 
or,  since  F+  \  a'  =  \  (F+  F'\ 

F^  =  J  (**  +  *")  -  i  [60  -  A  [d0  -  A  C/0  -  &c.)]]          (73) 

which  is  known  as  the  formula  for  interpolating  into  the  middle. 

When  the  third  differences  are  constant,  d^  /0,  &c.  are  zero, 
and  the  rule  for  interpolating  into  the  middle  between  two  func- 
tions is  simply  :  From  the  mean  of  the  two  functions  subtract  one- 
eighth  the  mean  of  the  second  differences  which  stand  against  the  func- 
tions. Interpolation  by  this  rule  is  correct  to  third  differences 
inclusive. 

The  formula  (73)  is  especially  convenient  in  computing  tables. 
Values  of  the  function  to  be  tabulated  are  directly  computed  for 
values  of  the  argument  differing  by  2mw  ;  then  interpolating  a 
value  into  the  middle  between  each  two  of  these,  the  arguments 
now  differ  by  2m~lw  ;  again  interpolating  into  the  middle  between 
each  two  of  the  resulting  series,  we  obtain  a  series  with  argu- 
ments differing  by  2wl~2w  ;  and  so  on,  until  the  interval  of  the 
argument  is  reduced  to  2>n~mw  or  w. 

EXAMPLE.  —  Find  the  moon's  right  ascension  for  1856  March 
5,  6*,  from  the  values  of  the  Ephemeris  for  noon  and  midnight. 

This  is  the  same  as  the  example  of  Art.  69  ;  but,  as  6A  is  the 
middle  instant  between  noon  and  midnight,  the  result  will  be 
obtained  by  the  formula  (73)  in  the  following  simple  manner. 
We  have  from  the  table  in  Art.  69 


b0  =  —  38U6 

9,        -73g^o^—   0.52          38.68  Xj=  +  4  .83 

-38.68  F&>=22   12   56.74 

73,  In  case  we  have  to  interpolate  between  the  last  two  values 
of  a  given  series,  we  may  consider  the  series  in  inverse  order, 
the  arguments  being  T,  T—w,  T—2w,  &c.,  T  being  the  last 
argument.  The  signs  of  the  odd  differences  will  then  be  changed, 
and,  taking  the  last  differences  in  the  several  columns  as  a,  6,  <?,  d, 
&c.,  the  interpolation  will  be  effected  by  (68). 


INTERPOLATION.  89 

74.  The  interpolation  formulas  arranged  according  to  the  powers  of 
the  fractional  part  of  the  argument. 

When  several  values  of  the  function  are  to  be  inserted  between 
two  of  the  given  series,  -it  is  often  convenient  to  employ  the 
formula  arranged  according  to  the  powers  of  TU  Performing  the 
multiplications  of  the  factors  indicated  in  (68),  and  arranging  the 
terms,  we  obtain 

ITWr^.F+nCrt  —  Jft-fic  —  Jd  +  Jc  —  &c.) 


+  &c  .........  (74) 

where  the  differences  are  obtained  according  to  the  schedule  in 
Art.  67. 

Transforming  (71)  in  the  same  manner,  we  have 

^»>  =  F  -f-  n  (a  —  i  c  -f  g15  e  —  &c.) 


_(rf_&c.) 

1.2.3.4V 

w5  p    , 

(e  —  Ac.) 


-f  Ac.  .  (75) 

where  the  differences  a,  c,  e,  are  the  mean  interpolated  odd  dif- 
ferences in  the  line  of  the  function  F  of  the  schedule  Art.  69. 

75.  Derivatives  of  a  tabulated  function. — When  the  analytical  ex- 
pression of  a  function  is  given,  its  derivatives  may  be  directly 
found  by  successive  differentiation  ;  but  when  this  expression  is 
not  known,  or  when  it  is  very  complicated,  we  may  obtain  values 
of  the  derivatives,  for  particular  values  of  the  variable,  from  the 
tabulated  values  of  the  functions  by  means  of  their  differences. 

Denoting  the  argument  by  T  +  nw,  its  corresponding  function 


90  INTERPOLATION. 

by/(2'-|-  nw},  the  successive  derivatives  of  this  function  cor- 
responding to  the  same  value  of  the  argument  will  be  denoted 
by  f'(T+nw\  f"(T+nw),  f"'(T  +  nw\  &c.,  and  f(T), 
f'(T),  f"(T),  £c.,  will  denote  the  values  of  the  function  and 
its  derivatives  corresponding  to  the  argument  T,  or  when  n  =  0. 
Hence,  if  we  regard  nw  as  the  variable,  we  shall  have,  by  Mac- 
laurin's  Theorem, 

f(T  +  nw)  =  /(T)  +f'(T)  nw  +  /''(T)!p|  +  Ac. 

Comparing  the  coefficients  of  the  several  powers  of  n  in  this 
formula  with  those  in  (74),  we  have 

f'(T)  =  —  (a  -  }  b  +  i  c  -  \d  +  i  e  -  &c.) 
w; 

/'(T)  =-1  (6  -  c  +  11  d  -  |  e  +  &c.) 


<«  -&c.) 
&c.  &c  .......  (76) 

the  differences  being  taken  as  in  Art.  67. 

Still  more  convenient    expressions  are  found  by  comparing 
Maclaurin's  Theorem  with  (75);  namely: 

/'(T)  =  JL  (a  -  64  c  +  A  e  -  &c.) 


&c.  &c.  (77) 

the  differences  being  found  according  to  the  schedule  in  Art.  69, 
and  the  odd  differences,  «,  e,  e,  &c.,  being  interpolated  means. 


STAR    CATALOGUES.  91 

The  preceding  formulae  determine  the  derivatives  for  the  value 
Z'of  the  argument.  To  find  them  for  any  other  value,  we  have, 
by  differentiating  Maclaurin's  Formula  with  reference  to  nw, 


f'(T+nw)  =  f'(T)  +f"(T)  .  nw  +  if'"(T).nW  +  &c.       (78) 

in  which  we  may  substitute  the  values  of  f(T),f'(T),  £c.  from 
(76)  or  (77). 

In  like  manner,  by  successive  differentiations  of  (78)  we  ob- 
tain 


w)  =f"  (T)  +  /'"(T).  nw  -j-  */*  (T).  n*w*  +  &c. 
/'"  (T-f  nW)=f"(T)  -|-/iv  (T).  nw  +  &c. 
&c.  &c. 

76.  An  immediate  application  of  (76)  or  (77)  is   the  compu- 
tation of  the  differences  in  a  unit  of  time  of  the  functions  in  the 
Ephemeris;    for  this  difference  is  nothing  more  than  the  first 
derivative,  denoted  above  by  the  symbol  /'. 

EXAMPLE.  —  Find  the  difference  of  the  moon's  right  ascension 
in  one  minute  for  1856  March  5,  0*. 

We  have  in  Art.  70,  for  T  =  March  5,  0A,  a  =  29™  6'.71, 
e  =  -f  O.79,  e  =  —  Os.54,  and  w  =  12*  =  720*.  Hence,  by  the 
first  equation  of  (77), 

f(T)  =  ,i0  (29-  6'.71  —  0-.13  —  0'.02)  =  2-.425S 

On  interpolation,  consult  also  ENCKE  in  the  Jahrbuch  for  1830 
and  1837. 

STAR   CATALOGUES. 

77.  The  Nautical  Almanac  gives  the  position  of  only  a  small 
number  of  stars.     The  positions  of  others  are  to  be  found  in 
the  Catalogues  of  stars.      These  are  lists  of  stars   arranged   in 
the  order  of  their  right  ascensions,  with  the  data  from  which 
their  apparent  right   ascensions  and  declinations  may  be  ob- 
tained for  any  given  date. 

The  right  ascension  and  declination  of  the  so-called  fixed 
stars  are,  in  fact,  ever  changing:  1st,  by  precession,  nutation, 
and  aberration  (hereafter  to  be  specially  treated  of),  which  are 
not  changes  in  the  absolute  position  of  the  stars,  but  are  either 
changes  in  the  circles  to  which  the  stars  are  referred  by  sphe- 
rical co-ordinates  (precession  and  nutation),  or  apparent  changes 
arising  from  the  observer's  motion  (aberration);  2d,  by  the 


92  STAR    CATALOGUES. 

proper  motion  of  the  stars  themselves,  which  is  a  real  change  of 
the  star's  absolute  position. 

In  the  catalogues,  the  stars  are  referred  to  a  mean  equator 
and  a  mean  equinox  at  some  assumed  epoch.  The  place  of  a 
star  so  referred  at  any  time  is  called  its  mean  place  at  that  time ; 
that  of  a  star  referred  to  the  true  equator  and  true  equinox, 
its  true  place ;  that  in  which  the  star  appears  to  the  observer  in 
motion,  its  apparent  place.  The  mean  place  at  any  time  will  be 
found  from  that  of  the  catalogue  simply  by  applying  the  preces- 
sion and  the  proper  motion  for  the  interval  of  time  from  the 
epoch  of  the  catalogue.  The  true  place  will  then  be  found  by 
correcting  the  mean  place  for  nutation ;  and  finally  the  appa- 
rent place  will  be  found  by  correcting  the  true  place  for  aber- 
ration. 

To  facilitate  the  application  of  these  corrections,  BESSEL  pro- 
posed the  following  very  simple  arrangement.  He  showed 
that  if 

a0,  80=  the  star's  mean  right  asc.  and  dec.  at  the  beginning  of  the 

year, 

a,  3  =  the  apparent  right  asc.  and  dec.  at  a  time  T  of  that  year, 
r  =  the  time  from  the  beginning  of  the  year  expressed  in  decimal 

parts  of  a  year, 

ft,  ft?  —  the  annual  proper  motion  of  the  star  in  right  asc.  and  dec. 
respectively, 

then, 

a  =  a0  -f  TH  +  Aa  +  Bb  +  Cc  +  Dd  +  E      \ 
d  =  d0+rn'+  Aa'  +  Bb'  +  Cc1  +  Dd'  / 

in  which  «,  6,  c,  rf,  a',  6',  c',  d'  are  functions  of  the  star's  right 
ascension  and  declination,  and  may,  therefore,  be  computed  for 
each  star  and  given  with  it  in  the  catalogue ;  A,  B,  C,  D,  E 
are  functions  of  the  sun's  longitude,  the  moon's  longitude,  the 
longitude  of  the  moon's  ascending  node,  and  the  obliquity  of  the 
ecliptic,  all  of  which  depend  on  the  time,  so  that  A,  B,  C,  D,  E 
may  be  regarded  simply  as  functions  of  the  time,  and  given  in 
the  Nautical  Almanac  for  the  given  year  and  day;  E  is  a 
very  small  correction,  usually  neglected,  as  it  can  never  ex- 
ceed 0".05. 

If  the  catalogue  does  not  give  the  constants  <z,  6,  <?,  d,  a',  b',  c', 
dr,  they  may  be  computed,  for  the  year  1850,  by  the  following 
formula?  (see  Chap.  XL  p.  648): 


STAR    CATALOGUES.  93 

a  -=  46".077  -f  20".056  sin  a  tan  3  a'  =  20".056  cos  a 

b  =  cos  a  tan  3  b'  =  —  sin  a 

c  —  cos  a  sec  3  (f  —  tan  e  coa  d  —  sin  a  sin  d 

d  =  sin  a  sec  3  d'  =  cos  *  sin  <5 

in  which  e  =  obliquity  of  the  ecliptic.  Or  we  may  resort  to 
what  are  usually  called  the  independent  constants,  and  dispense 
with  the  <?,  b,  c,  d,  a',  b',  c',  d'  altogether,  proceeding  then  by 
the  formula 

a  =  «0  -f-  r,u  -f/          -f  g  sin  (67  -f-  «)  tan  d  -f-  A  sin  (JT-f  «)  sec  <5 1 
£  =  <50+  r//-f  i  cos  «5  -fry  cos (£  +  a)  -f  A  cos  (Jf-f  a)  sin  d  }  < 

the  independent  constants  /,  g,  G,  A,  II,  £  being  given  in  the 
Ephemeris,  together  with  the  value  of  r  for  the  given  date, 
expressed  decimally. 

It  should  be  observed  that  the  constants  a,  b,  c,  d,  a',  b',  e' ,  d' 
are  not  absolutely  constant,  since  they  depend  on  the  right 
ascension  and  declination,  which  are  slowly  changing:  unless, 
therefore,  the  catalogue  which  contains  them  gives  also  their 
variations,  or  unless  the  time  to  which  we  wish  to  reduce  is  not 
very  remote  from  the  epoch  of  the  catalogue,  it  may  be  prefer- 
able to  use  the  independent  constants. 

In  forming  the  products  Aa>  Bb,  &c.,  attention  must  of  course 
be  paid  to  the  algebraic  signs  of  the  factors.  The  signs  of  A,  B, 
(7,  D  are,  in  the  Ephemerides,  prefixed  to  their  logarithms  *  and 
the  signs  of  a,  6,  c,  &c.  are  in  some  catalogues  (as  that  of  the 
British  Association)  also  prefixed  to  their  logarithms ;  but  I 
shall  here,  as  elsewhere  in  this  work,  mark  only  the  logarithms 
of  negative  factors,  prefixing  to  them  the  letter  n. 

It  should  be   remarked,  also,  that  the  B.   A.   C.*  gives  the 

*  B.  A.  C. — British  Association  Catalogue,  containing  8377  stars,  distributed  in  all 
parts  of  the  heavens ;  a  very  useful  work,  but  not  of  the  highest  degree  of  precision. 
The  Greenwich  Catalogues,  published  from  time  to  time,  are  more  reliable,  though 
less  comprehensive.  For  the  places  of  certain  fundamental  stars,  see  BESS  EL'S 
Tabulte  Rcgiomontanx  and  its  continuation  by  WOLFERS  and  ZECH. 

LALANDE'S  Ilistoire  Celeste  contains  nearly  50,000  stars,  most  of  which  are  em- 
braced in  a  catalogue  published  by  the  British  Association,  reduced,  under  the 
direction  of  F.  Baily,  from  the  original  work  of  Lalande.  The  Konigsberg  Observa- 
tions embrace  the  series  known  as  BESSEL'S  ZONES,  the  most  extensive  series  of 
observations  of  small  stars  yet  published.  The  original  observations  are  given  with 
data  for  their  reduction,  but  an  important  part  of  them  is  given  in  WEISSE'S  Posi- 
tiones  Mediae  StMarum  fixarum  in  Zonis  Regiomontanis  a  BESSELIO  inter — 15°  ft-J-150 
declin.  observat.,  containing  nearly  32,000  stars. 

See  also  STRUVE'S  Catat.  general**,  and  the  catalogues  of  ARGELANDER,  RUMKER, 


94 


STAR    CATALOGUES. 


north  polar  distance  instead  of  the  declination,  or  TTO—  90°  —  r?0; 
and,  since  x  decreases  when  3  increases,  the  corrections  change 
their  sign.  This  has  been  provided  for  by  changing  the  signs  of 
//,  a',  b',  c1',  d'  in  the  catalogue  itself.  Moreover,  in  this  cata- 
logue, a,  6,  a',  bf  denote  BESSEL'S  c,  d,  c',  rf',  and  vice  versa ;  and 
to  correspond  with  this,  the  J.,  B,  (7,  D  of  the  British  Almanac 
denote  BESSEL'S  (7,  -D,  A,  B.  The  same  inversion  also  exists  in 
the  American  Ephemeris  prior  to  the  year  1865,  but  in  the  volume 
for  1865  the  original  notation  is  restored. 

EXAMPLE. — Find  the  apparent  right  ascension  and  declination 
of  a  Tauri  for  June  15,  1865,  from  Argelander's  Catalogue. 
This  star  is  Argel.  108 ;  whence  we  take  for 


Jan.  1,  1830.     Mean  R.  A.  =  4*  26"'  10«.43 


Ann.  prec.      =  +  3«.  428 
Prop,  motion  —  -f  0  .  005 


g5 


Mean  Decl.  =  -f  16°    9'  36".  0 
4-  7".90 


=  +    2 

0.155 

= 

-f      4  30  .55 

Jan.  1, 

1865,  a0=4    28 

10.585 

*o  =  + 

16    14     6  .55 

We  next  take  the  logarithms 

from  the  Catal. 

logs,     a  0.5352 

b     7.8794 

c     8.4329 

d    8.8058 

from  Am.  Ephem.  ) 
for  June  15,  1865,  / 

logs.    A  9.7877 

B    0.9437 

CnO.2125 

D  nl.3089 

from  the  Catal. 

logs,      a'  0.8934 

b'  n9.9607 

c'    9.2019 

d'    9.0378 

logs.  Aa  0.3229 

Bb     8.8231 

Cc  n8.645t 

/></  «0.1147 

logs.  Aa'  0.6811 

Bb'  nO.9044 

Cc'  n9.4144 

Dd'  nO.3467 

Corr.  of  a0,  A  a  = 

-f  2M03,     Bb  = 

-f  0».067,     Cc 

—  _  0«.044,     D 

•rf  ==_1».302 

Corr.  of  <L,  Aa'  = 

+  4".  80,      Bb'  = 

—  8".02,      Cc' 

—  —  0".26,      L 

>d'  —  —2".  22 

We  have  also  from  the  catalogue  p  =  -f-  0*.005,  //'  =   -  0".17. 
The  fraction  of  a  year  for  June  15,  1865,  is  r  =  0.46 ;  and  hence 


Jan.  1,  1865,  a0  =  4*  28™  1C'. 685 

Sum  of  corr.  of  a0       =       -j-       0  .824 

rfi  =       -f-       0  .002 

June  15,  1865  a  =  4   28     11  .411 


rJ0=  -f  16°  14'  6".55 

Sum  of  corr.  of  (50  =  -  5  .70 

TU'  =  —  0  .08 

f5=r  +  16    14  0  .77 


78.  When  the  greatest  precision  is  required,  we  should  con- 
sider the  change  in  the  star's  place  even  in  a  fraction  of  a  day, 
and  therefore  also  the  change  while  the  star  is  passing  from  one 
meridian  to  another;  also  the  secular  variation  and  the  changes 


PIAZZI,  SANTINI  ;  and  the  published  observations  of  the  principal  observatories, 
also  a  list  of  catalogues  in  the  introduction  to  the  B.  A.  C. 


THE    EARTH.  95 

in  the  precession  and  in  the  logarithms  of  the  constants.  Fur- 
ther, it  is  to  be  observed  that  the  annual  precession  of  the  cata- 
logues is  for  a  mean  year  of  365d  5*. 8.  But  for  a  fuller  consider- 
ation of  this  subject  see  Chapter  XI. 


CHAPTER    III. 

FIGURE   AND   DIMENSIONS  OF   THE  EARTH. 

79.  THE  apparent  positions  of  those  heavenly  bodies  which  are 
within  measurable  distances  from  the  earth  are  different  for  ob- 
servers on  different  parts  of  the  earth's  surface,  and,  therefore, 
before  we  can  compare  observations  taken  in  different  places  we 
must  have  some  knowledge  of  the  form  and  dimensions  of  the 
earth.     I  must  refer  the   reader   to   geodetical  works   for  the 
methods  by  which  the  exact  dimensions  of  the  earth  have  been 
obtained,  and  shall  here  assume  such  of  the  results  as  I  shall 
have  occasion  hereafter  to  apply. 

The  figure  of  the  earth  is  very  nearly  that  of  an  oblate  spheroid, 
that  is,  an  ellipsoid  generated  by  the  revolution  of  an  ellipse 
about  its  minor  axis.  The  section  made  by  a  plane  through  the 
earth's  axis  is  nearly  an  ellipse,  of  which  the  major  axis  is  the 
equatorial  and  the  minor  axis  the  polar  diameter  of  the  earth. 
Accurate  geodetical  measurements  have  shown  that  there  are 
small  deviations  from  the  regular  ellipsoid ;  but  it  is  sufficient 
for  the  purposes  of  astronomy  to  assume  all  the  meridians  to  be 
ellipses  with  the  mean  dimensions  deduced  from  all  the  measures 
made  in  various  parts  of  the  earth. 

80.  Let  EPQP',  Fig.  11,  be  one  of  the  elliptical  meridians  of 
the   earth,  JEQ  the   diameter   of  the   equator,  PPf   the   polar 
diameter,  or  axis  of  the  earth,  0  the  centre,  F  a  focus  of  the 
ellipse.    Let 

a  =  the  semi-major  axis,  or  equatorial  radius,  =  CE, 
b  =  the  semi-minor  axis,  or  polar  radius,          =  CP, 
c  =  the  compression  of  the  earth, 
e  =  the  eccentricity  of  the  meridian. 


96 


REDUCTION    OF   LATITUDE. 


By  the  compression  is  meant  the  difference  of  the  equatorial 

and  polar  radii  expressed  in  parts 
of  the  equatorial  radius  as  unity,  or 


a  —  b  b 

A     ___    _  .     1       _  , 

a  a 


The  eccentricity  of  the  meridian  is 
E>    the  distance  of  either  focus  from 
the  centre,  also  expressed  in  parts 
of   the   equatorial   radius,   or,   in 
Fig.  11, 


e  s±= 


CF 
OE 


But,  since  PF=  CE,  we  have, 


PC* 


that  is, 


CE' 


or 


(81) 


By  a  combination  of  all  the  most  reliable  measures,  BESSEL 
deduced  the  most  probable  form  of  the  spheroid,  or  that  which 
most  nearly  represents  all  the  observations  that  have  been  made 
in  different  parts  of  the  world.  He  found* 


b^  _  298.1523 

a  ~  ~  299.1528 


or 


whence,  by  (81), 

log  e  =  8.912205 


299.1528 
e  ==  .0816967 


—  ee)  =  9.9985458 


*  Astronomische  Nachrichten,  No.  438.     See  also  Encke's  Tablei  of  the  dimensions 
of  the  terrestrial  spheroid  in  the  Jahrbuch  for  1852. 


REDUCTION    OF    LATITUDE.  97 

The  absolute  lengths  of  the  semi-axes,  according  to  BESSEL,  are, 

a  =  6377397.15  metres  =  6974532.34  yds.  =  3962.802  miles 
b  =  6356078.96       "       =  6951218.06    «     =  3949.555     « 

81.  To  find  the  reduction  of  the  latitude  for  the  compression  of  the 
earth. 

Let  A,  Fig.  11,  be  a  point  on  the  surface  of  the  earth;  AT  the 
tangent  to  the  meridian  at  that  point ;  A  0,  perpendicular  to  A  T, 
the  normal  to  the  earth's  surface  at  A.  A  plane  touching  the 
earth's  surface  at  A  is  the  plane  of  the  horizon  at  that  point 
(Art.  3),  and  therefore  A  0,  which  is  perpendicular  to  that  plane, 
represents  the  vertical  line  of  the  observer  at  A.  This  vertical 
line  does  not  coincide  with  the  radius,  except  at  the  equator  and 
the  poles.  If  we  produce  CE,  OA,  and  CA  to  meet  the  celestial 
sphere  in  Er,  Z,  and  Zf  respectively,  the  angle  ZO'E'  is  the 
declination  of  the  zenith,  or  (Art.  7)  the  geographical  latitude,  and 
Z  \s  the  geographical  zenith ;  the  angle  Z'CE'  is  the  declination 
of  the  geocentric  zenith  Z',  and  is  called  the  geocentric  or  reduced 
latitude;  and  ZAZ'  =  CAO  is  called  the  reduction  of  the  latitude. 
It  is  evident  that  the  geocentric  is  always  less  than  the  geogra- 
phical latitude. 

Now,  if  we  take  the  axes  of  the  ellipse  as  the  axes  of  co-ordi- 
nates, the  centre  being  the  origin,  and  denote  by  x  the  abscissa, 
and  by  y  the  ordinate  of  any  point  of  the  curve,  by  a  and  b  the 
semi-major  and  semi-minor  axes  respectively,  the  equation  of 
the  ellipse  is 


If  we  put 

<p  =  the  geographical  latitude, 
</>'  =  the  geocentric  " 

we  have,  since  (p  is  the  angle  which  the  normal  makes  with  the 
axis  of  abscissae, 

dx 
tan  <p  = 

and  from  the  triangle  A  CjB, 

v 

tan  /  =  — 
x 

VOL.  I.— 7 


98  REDUCTION    OF   LATITUDE. 

Differentiating  the  equation  of  the  ellipse,  we  have 

y  _         b2    dx 

x  a2    dy 

or 

62 

tan  <p'  —  _  tan  <p  —  (1  —  e2)  tan  <p  (82) 

a2 

which  determines  the  relation  between  <p  and  ^>'. 

To  find  the  difference  ip  —  <p',  or  the  reduction  of  the  latitude, 
we  have  recourse  to  the  general  development  in  series  of  an 
equation  of  the  form 

tan  x  =  p  tan  y 
which  [PL  Trig.  Art.  254]  is 

x  —  y  —  q  sin  2y  -f  $  <£  sin  4y  -}-  &c. 
in  which 

y-i 
,-t-i 

Applying  this  to  the  development  of  (82),  we  find,  after  divid- 
ing by  sin  V  to  reduce  the  terms  of  the  series  to  seconds, 

1  -  <?'  =  -  -Ti-f  sin  2?  -  — ^—  sin  4?  -  &c.  (83) 

sm  1"  2  sin  1" 

in  which 

_p  —  1  _    1  —  e*  —  I  _  e* 

q  ~ p  +  1  "  1  —  e2  -f  1  "     ~  2— e2 
Employing  BESSEL'S  value  of  e,  we  find 

^—  =  69CT65 ^ =  —  1".16 

sin  1"  2  sin  1" 

and,  the  subsequent  terms  being  insensible, 

<p  —  <?'  =  690".65  sin  2?  —  1".16  sin  4^»  (83*) 

by  which  <p  —  <pr  is  readily  computed  for  given  values  of  <p.  Its 
value  will  be  found  in  our  Table  III.  Vol.  II.  for  any  given 
value  of  (p. 

EXAMPLE. — Find  the  reduced  latitude  when  <p  =  35°.  "We  find 
by  (83),  or  Table  III., 

^  —  <f  =  648".25  =  10'  48".25 
and  hence  the  reduced  or  geocentric  latitude 
<?'  =  34°  49'  11".75 


RADIUS    OF   THE    EAETH.  99 

82.   To  find  the  radius  of  the  terrestrial  spheroid  for  a  given  latitude. 
Let 

p  =  the  radius  for  the  latitude  <p  =  AC. 
We  have 


=  /  a? 


To  express  x  and  y  in  terms  of  y,  we  have  from  the  equation  of 
the  ellipse  and  its  differential  equation,  after  substituting  1  —  e2 

f  b* 

for  —5 
a2 


=  a2 


1—  e2 

^  =  (1  —  e*~)  tan  9 
oc 

from  which  by  a  simple  elimination  we  find 

a  cos   > 

' 


—  e2  sin2?) 
(1  —  e2)  a  siny» 
=  l/(l—€«  sinV) 
and  hence 

=         /ri-2^siny-h^BinVI 
\L  1—  e2sinV  J 


by  which  the  value  of  p  may  be  computed.     The  logarithm  of 
p,  putting  a  =  1,  is  given  in  our  Table  III.  Vol.  II. 

But  the  logarithm  of  p  may  be  more  conveniently  found  by  a 
series.     If  in  (84)  we  substitute 


=  l  —f* 
=  i  (1  —  cos 


we  find,  putting  a  =  1, 


Kow  (PI.  Trig.  Art.  260)  if  we  have  an  expression  of  the  form 
X  =  |/(1  -1-  ro'  —  2m  cos  C) 


100  RADIUS  OF  THE  EARTH. 

we  have,  if  M  —  the  modulus  of  the  common  system  of  loga- 
rithms, 

™l  n   .    m'  cos  2C   ,    m*  cos  3(?   ,          \ 

log  X  =  —  M  (  m  cos  C  -\  --  -  --  1  ---  -f  &c.  1      (J5) 

by  which  we  may  develop  the  logarithms  of  the  numerator  and 
denominator  of  the  above  radical. 
Hence  we  find 

_j_  f  2  /  77j2  _  77/2 

'  —  cos  4> 


1  _j_  f  2       / 

log  p  =  log  --  -  —  f-  M  I  (m  —  m')  cos  2^ 
1  -\-  f     \ 


-j —  -  cos  6y>  —  &c.   I 


in  which  we  have  put  for  brevity 

I-/2  I-/ 


m  = 

1-f/2 

Restoring  the  value    of  /  =  |/(1  —  e2)  and  computing  the 
numerical  values  of  the  coefficients,  we  find 

log  p  =  9.9992747  +  0.0007271  cos  2^  —  0.0000018  cos  4?      (85) 

as  given  by  ENCKE  in  the  Jahrbuch  for  1852. 

The  values  of  p  and  <pf  may  also  be  determined  under  another 
form  which  will  hereafter  be  found  useful. 

We  have  in  Fig.  11,  p  sin  y'  =  y,  p  cos  <pf  —  x9  or 


a  (I  —  e2)  sin  <p 
p  sin  <p'  = 


.  a  cos  <p 

O  COS  v  = 


(86) 


which  may  be  put  under  a  simple  form  by  introducing  an  auxi- 
liary 4>>  so  that 

sin  4  =  e  sin  <p  ^ 

p  sin  <f'  =  a  (1  —  e2)  sin  <f>  sec  4*  V      (87) 

p  cos  <p'  =  a  cos  #>  sec  4*  J 

We  can  also  deduce  from  these, 

P  sin  (<p  —  <f>'}  —  £  «e2  sin  2^>  sec  4/  1 

x  /\  r      (88) 

/o  cos  (<f>  —  <p)  =  a  cos  4  J 


NORMAL.  101 

Hence,  also,  the  following: 


83.  To  find  the  length  of  the  normal  terminating  in  the  axis,  for  a 
given  latitude. 

Putting 

N=  the  normal  =  AO  (Fig.  11), 

we  have  evidently 

N=!L™iL  =  --  «  - 

cos  </>  |/(1  —  el  sin2  ^) 

or,  employing  the  auxiliary  ^  of  the  preceding  article, 

N  =•  a  sec  4 

84.  Jb  jtfnrf  the  distance  from  the  centre  to  the  intersection  of  the 
normal  with  the  axis. 

Denoting  this  distance  by  ai  (so  that  i  denotes  the  distance 
when  a  =  1),  we  have  in  Fig.  11, 

ai  =  CO 
and,  from  the  triangle  A  CO, 

ai  =  p  8in  ^  ~  ?') 

COS  <p 

or,  by  (88), 

ae*  sin  a> 

-'t     =  <M'"''y'ec* 


85.  Jb  ^n<^  </ie  radius  of  curvature  of  the  terrestrial  meridian  for  a 
given  latitude.  —  Denoting  this  radius  by  R,  we  have,  from  the  dif- 
ferential calculus, 


where  we  employ  the  notation  D^j/,  D/  i/  to  denote  the  first 
and  second  differential  coefficients  of  y  relatively  to  x.  We 
have  from  the  equation  of  the  ellipse 


off 


102  DEVIATIONS    OF  THE    PLUMB    LINE. 

whence 

E  :==  ~ 777 

Observing  that  b2  =  a2  (1  —  e2),  we  find,  by  substituting  the  values 
of  x  and  y  in  terms  of  tp  (p.  99), 

T>  •TV         ^  /  /oo\ 

H=- a — r  (y^) 

(1  —  e2  sin2  p)f 

EXAMPLE. — Find  the  radius  of  curvature  for  the  latitude  of 
Greenwich,  tp  =  51°  28'  38".2,  taking  a  =  6377397  metres.  We 

find 

E  =  6373850  metres. 

86.  Abnormal  deviations  of  the  plumb  line. — Granting  the  geo- 
metrical figure  of  the  earth  to  be  that  of  an  ellipsoid  of  revolu- 
tion whose  dimensions,  taking  the  mean  level  of  the  sea,  are  as 
given  in  Art.  80,  it  must  not  be  inferred  that  the  direction  of  the 
plumb  line  at  any  point  of  the  surface  always  coincides  precisely 
with  the  normal  of  the  ellipsoid.  It  would  do  so,  indeed,  if  the 
earth  were  an  exact  ellipsoid  composed  of  perfectly  homoge- 
neous matter,  or  if,  originally  homogeneous  and  plastic,  it  has 
assumed  its  present  form  solely  under  the  influence  of  the 
attraction  of  gravitation  combined  with  the  rotation  on  its  axis. 
But  experience  has  shown*  that  the  plumb  line  mostly  deviates 
from  the  normal  to  the  regular  ellipsoid,  not  only  towards  the 
north  or  south,  but  also  towards  the  east  or  west;  so  that  the 
apparent  zenith  as  indicated  by  the  plumb  line  differs  from  the 
true  zenith  corresponding  to  the  normal  both  in  declination  and 
right  ascension.  These  deviations  are  due  to  local  irregularities 
both  in  the  figure  and  the  density  of  the  earth.  Their  amount  is, 
however,  very  small,  seldom  reaching  more  than  3"  of  arc  in 
any  direction. 

In  order  to  eliminate  the  influence  of  these  deviations  at  a 
given  place,  observations  are  made  at  a  number  of  places  as 
nearly  as  possible  symmetrically  situated  around  it,  and,  as- 
suming the  dimensions  of  the  general  ellipsoid  to  be  as  we  have 
given  them,  the  direction  of  the  plumb  line  at  the  given  place  is 
deduced  from  its  direction  at  each  of  the  assumed  places  (by 

*  U.S.  Coast  Survey  Report  fof  1853,  p.  14*. 


REDUCTION  TO  THE  CENTRE  OF  THE  EARTH.        103 

the  aid  of  the  geodetic  measures  of  its  distance  and  direction 
from  each) ;  or,  which  is  the  same  thing,  the  latitude  and  longi- 
tude of  the  place  are  deduced  from  those  of  each  *>f  the  assumed 
places :  then  the  mean  of  all  the  resulting  latitudes  is  the  geodetic 
latitude,  and  the  mean  of  all  the  resulting  longitudes  is  the  geodetic 
longitude,  of  the  place.  These  quantities,  then,  correspond  as 
nearly  as  possible  to  the  true  normal  of  the  regular  ellipsoid ; 
the  geodetic  latitude  being  the  angle  which  this  normal  makes 
with  the  plane  of  the  equator,  and  the  geodetic  longitude  being 
the  angle  which  the  meridian  plane  containing  this  normal 
makes  with  the  plane  of  the  first  meridian.  The  geodetic  lati- 
tude is  identical  with  the  geographical  latitude  as  we  have  defined 
it  in  Art.  81. 

The  astronomical  latitude  of  a  place  is  the  declination  of  the 
apparent  zenith  indicated  by  the  actual  plumb  line ;  but,  unless 
wrhen  the  contrary  is  stated,  it  will  be  hereafter  understood  to  be 
identical  with  the  geographical  or  geodetic  latitude. 

It  has  recently  been  attempted  to  show  that  the  earth  differs 
sensibly  from  an  ellipsoid  of  revolution;*  but  no  deduction  of 
this  kind  can  be  safely  made  until  the  anomalous  deviations  of 
the  plumb  line  above  noticed  have  been  eliminated  from  the 
discussion. 


CHAPTER    IV. 

REDUCTION   OF   OBSERVATIONS   TO   THE   CENTRE   OF   THE   EARTH. 

87.  THE  places  of  stars  given  in  the  Ephemerides  are  those  in 
which  the  stars  would  be  seen  by  an  observer  at  the  centre  of 
the  earth,  and  are  called  geocentric,  or  true,  places.  Those  observed 
from  the  surface  of  the  earth  are  called  observed,  or  apparent, 
places. 

It  must  be  remarked,  however,  that  the  geocentric  places  of 
the  Ephemeris  are  also  called  apparent  places  when  it  is  intended 

*  See  Astr.  Nach.  No.  1303. 


104  PARALLAX. 

to  distinguish  them  from  mean  places,  a  distinction  which  will 
be  considered  hereafter  (Chap.  XI.). 

It  will  also  be  noticed  that  we  frequently  use  the  terms  true 
and  apparent  as  relative  terms  only;  as,  for  example,  in  treating 
of  the  effect  of  parallax,  the  place  of  a  star  as  seen  from  the 
centre  of  the  earth  may  be  called  true,  and  that  in  which  it 
would  be  seen  from  the  surface  of  the  earth  were  there  no 
atmosphere,  may  in  relation,  to  the  former  be  called  apparent; 
but  in  considering  the  effect  of  refraction,  the  star's  place  as  it 
would  be  seen  from  the  surface  of  the  earth  were  there  no  atmo- 
sphere may  be  called  true,  and  the  place  as  affected  by  the  re- 
fraction may  in  relation  to  the  former  be  called  apparent;  and 
similarly  in  other  cases. 

PARALLAX. 

88.  The  parallax  of  a  star  is,  in  general,  the  difference  of  the 
directions  of  the  straight  lines  drawn  to  the  star  from  two  different 
points.  The  difference  of  direction  of  two  straight  lines  being 
simply  the  angle  contained  between  them,  we  may  also  define 
parallax  as  the  angle  at  the  star  contained  by  the  lines  drawn  to 
the  two  points  from  which  it  is  supposed  to  be  viewed. 

In  astronomy  we  frequently  use  the  term  parallax  to  express 
the  difference  of  altitude  or  of  zenith  distance  of  a  star  seen 
from  the  surface  and  the  centre  of  the  earth  respectively; 
and,  in  order  to  express  parallax  in  respect  to  other  co-ordi- 
nates, proper  qualifying  terms  are  added,  as  "  parallax  in  decli- 
nation," &c. 

Assuming  (at  first)  the  earth  to  be  a  sphere,  let  A,  Fig.  12,  be 
the  position  of  the  observer  on  its  surface, 
C  the  centre,  CAZ  the  vertical  line,  and  S  a 
star  within  a  measurable  distance  CS  from 
the  centre.     All' ,  a  tangent -to  the  surface 
at  J.,  and  CH,  parallel  to  it,  drawn  through 
H,     the  centre,  may  each  be  regarded  as  lying 
in  the  plane  of  the  celestial  horizon  (note, 
p.  19).     The  true  or  geocentric  altitude  of 
the  star  above  the  celestial  horizon  is  then 
the  angle  8CH,  and  the  apparent  altitude  is 
the  angle  SAJP.     In  this  case  the  directions  of  the  star  from  C 
and  from  A  are  compared  with  each  other  by  referring  them  to  two 


PAKALLAX.  105 

lines  which  have  a  common  direction,  i.e.  parallel  lines.  But  a 
still  more  direct  method  of  comparison  is  obtained  by  referring 
them  to  one  and  the  same  straight  line,  as  CAZ,  Z  being  the 
zenith.  We  then  call  ZCS  the  true  and  ZAS  the  apparent 
zenith  distance,  and  these  are  evidently  the  complements  of  the 
true  and  apparent  altitudes  respectively. 
In  the  figure  we  have  at  once 

ZAS—  ZCS=ASC 

that  is,  the  parallax  in  zenith  distance  or  altitude  is  the  angle 
at  the  star  subtended  by  the  radius  of  the  earth.  When  the  star 
is  in  the  horizon,  as  at  Hf,  the  radius,  being  at  right  angles  to 
AH',  subtends  the  greatest  possible  angle  at  the  star  for  the  same 
distance,  and  this  maximum  angle  is  called  the  horizontal  parallax. 
The  equatorial  horizontal  parallax  of  a  star  is  the  maximum  angle 
subtended  at  the  star  by  the  equatorial  radius  of  the  earth. 

89.  To  find  the  equatorial  horizontal  parallax  of  a  star  at  a  given 
distance  from  the  centre  of  the  earth. 

Let 

TT  =  the  equatorial  horizontal  parallax, 

A  =  the  given  distance  of  the  star  from  the  earth's  centre, 

a  =  the  equatorial  radius  of  the  earth, 

we  have  from  the  triangle  CAR1  in  Fig.  12,  if  CA  is  the 
equatorial  radius, 

sin  TT  —  —  (93) 

J 

The  value  of  it  given  in  the  Ephemeris  is  always  that  which  is 
given  by  this  formula  when  for  J  we  employ  the  distance  of  the 
star  at  the  instant  for  which  the  parallax  is  given. 

90.  To  find  the  parallax  in  altitude  or  zenith  distance,  the  earth  being 
regarded  as  a  sphere. 

Let 

C  =  the  true  zenith  distance          =  ZCS  (Fig.  12), 
£'  —  the  apparent  zenith  distance  =  ZAS, 
p  =  the  parallax  in  alt.  or  z.  d.      =  CSA. 


106          .  PARALLAX. 

The    triangle    SAC  gives,   observing    that    the    angle   SAC 

=  180J  —  £', 

sin  p        a 

-r^  =  -  =  Bm  „ 

or, 

sinp  =  sin  (C'  —  C)  =  sin  n  sin  C'  (94) 

If  we  put 

A,  =  the  true  altitude, 

h'  =  the  apparent  altitude, 

then  it  follows  also  that 

smp  =  sin  (h  —  h')  =  sin  -  cos  hf  (95) 

Except  in  the  case  of  the  moon,  the  parallax  is  so  small  that  we 
may  consider  n  and  p  to  be  proportional  to  their  sines  [PL  Trig. 
Art.  55] ;  and  then  we  have 

p  =.  TT  sin  C'  =  n  cos  hf  (96) 

Since  when  f '  =  90°  we  have  sin  £'  =  1,  and  when  £'  =  0,  sin 
£'  —  0,  it  follows  that  the  parallax  is  a  maximum  when  the  star 
is  in  the  horizon,  and  zero  when  the  star  is  in  the  zenith. 

EXAMPLE. — Given  the  apparent  zenith  distance  of  Yenus, 
£'  =  64°  43',  and  the  horizontal  parallax  ^  =  20/r.O;  find  the 
geocentric  zenith  distance. 

log  TT  1.3010 

C'  =  64°  43'  0"0  log  sin  C'  9.9563 

p=  18.1  logp  L2573~ 

C  =  64  4241.9 

When  the  true  zenith  distance  is  given,  to  compute  the  paral- 
lax, we  may  first  use  this  true  zenith  distance  as  the  apparent, 
and  find  an  approximate  value  of  p  by  the  formula  p  =  TT  sin  £ ; 
then,  taking  the  approximate  value  of  £'  —  f  —  p,  we  compute  a 
more  exact  value  of  p  by  the  formula  (94)  or  (96).  This  second 
approximation  is  unnecessary  in  all  cases  except  that  of  the 
moon,  and  the  parallax  of  the  moon  is  so  great  that  it  becomes 
necessary  to  take  into  account  the  true  figure  of  the  earth,  as  in 
the  following  more  general  investigation  of  the  subject. 

91.  In  consequence  of  the  spheroidal  figure  of  the  earth,  the 
vertical  line  of  the  observer  does  not  pass  through  the  centre, 
and  therefore  the  geocentric  zenith  distance  cannot  be  directly 


PARALLAX.  107 

referred  to  this  line.  If,  however,  we  refer  it  to  the  radius  drawn 
from  the  place  of  observation  (or  CAZ1 ',  Fig.  11),  the  zenith  dis- 
tance is  that  measured  from  the  geocentric  zenith  of  the  place; 
whereas  it  is  desirable  to  use  the  geographical  zenith.  Hence 
we  shall  here  consider  the  geocentric  zenith  distance  to  be  the 
anorle  which  the  straight  line  drawn  from  the  centre  of  the  earth 

O  O 

to  the  star  makes  with  the  straight  line  drawn  through  the  centre 
of  the  earth  parallel  to  the  vertical  line  of  the  observer.  These  two 
vertical  lines  are  conceived  to  meet  the  celestial  sphere  in  the 
same  point,  namely,  the  geographical  zenith,  which  is  the 
common  vanishing  point  of  all  lines  perpendicular  to  the  plane 
of  the  horizon.  Thus  both  the  true  and  the  apparent  zenith 
distances  will  be  measured  upon  the  celestial  sphere  from  the 
pole  of  the  horizon. 

The  azimuth  of  a  star  is,  in  general,  the  angle  which  a  vertical 
plane  passing  through  the  star  makes  with  the  plane  of  the  meri- 
dian. When  such  a  vertical  plane  is  drawn  through  the  centre 
of  the  earth,  it  does  not  coincide  with  that  drawn  at  the  place  of 
observation,  since,  by  definition  (Art.  3),  the  vertical  plane  passes 
through  the  vertical  line,  and  the  vertical  lines  are  not  coincident. 
Hence  we  shall  have  to  consider  a  parallax  in  azimuth  as  well  as 
in  zenith  distance. 

92.  To  find  the  parallax  of  a  star  in  zenith  distance  and  azimuth 
when  the  geocentric  zenith  distance  and  azimuth  are  given,  and  the  earth 
is  regarded  as  a  spheroid.* 

Let  the  star  be  referred  to  three  co-ordinate  planes  at  right 
angles  to  each  other :  the  first,  the  plane  of  the  horizon  of  the 
observer;  the  second,  the  plane  of  the  meridian;  the  third,  the 
plane  of  the  prime  vertical.  Let  the  axis  of  x  be  the  meridian 
line,  or  intersection  of  the  plane  of  the  meridian  and  the  plane 
of  the  horizon ;  the  axis  of  y,  the  east  and  west  line  ;  the  axis 
of  2,  the  vertical  line.  Let  the  positive  axis  of  x  be  towards  the 
south;  the  positive  axis  of  ?/,  towards  the  west;  the  positive 
axis  of  2,  towards  the  zenith.  Let 

J'  =:  the  distance  of  the  star  from  the  origin,  which  is 

the  place  of  observation, 

C'  =  the  apparent  zenith  distance  of  the  star, 
A'  =  the  apparent  azimuth      "  "  " 

*  The  investigation  which  follows  is  nearly  the  same  as  that  of  OLBERS,  to  whom 
the  method  itself  is  due. 


108  PARALLAX. 

then,  x'  ?/',  z'  denoting  the  co-ordinates  of  the  star  in  this  system, 
we  have,  by  (39), 

x'  =  A'  sin  C'  cos  A' 
y'  =  A'  sin  C'  sin  A' 
z'  =  A'  cos  C' 

Again,  let  the  star  be  referred  by  rectangular  co-ordinates  to 
another  system  of  planes  parallel  to  the  former,  the  origin  now 
being  the  centre  of  the  earth.  In  the  celestial  sphere  these 
planes  still  represent  the  horizon,  the  meridian,  and  the  prime 
vertical.  If  then  in  this  system  we  put 

A  =  the  distance  of  the  star  from  the  origin, 
£  =  the  true  zenith  distance  of  the  star, 
A  =  the  true  azimuth  "         " 

and  denote  the  co-ordinates  of  the  star  in  this  system  by  x,  y, 
and  z,  we  have,  as  before, 

x  =  A  sin  C  cos  A 
y  —  A  sin  C  sin  A 

z  =  A  cos  C 

Now,  the  co-ordinates  of  the  place  of  observation  in  this  last 
system,  being  denoted  by  a,  6,  c,  we  have 

a  =  p  sin  (f  —  <f>')          b  —  0  c  —  p  cos  (y>  —  ^') 

in  which  p  —  the  earth's  radius  for  the  latitude  <p  of  the  place  of 
observation,  and  <p '  is  the  geocentric  latitude,  <p  —  <pf  being  the 
reduction  of  the  latitude,  Art.  81 ;  and  the  formulae  of  transforma- 
tion from  this  second  system  to  the  first  are  (Art.  33) 

x  —  x?  +  a          y  =y'  -\- b          z  =  z'  -}-  c 
or,  x'  =  x  —  a          y'  =  y  —  b          z'  =  z  —  c 

whence,  substituting  the  above  values  of  the  co-ordinates, 

A'  sin  £'  cos  A'  =  A  sin  C  cos  A  —  p  sin  (y  —  <?'}  "| 

A'  sin  C'  sin  A'  =  A  sin  C  sin  J.  >      (97) 

A'  cos  C'  =  A  cos  C  —  p  cos  (<?  —  <?') 

which  are  the  general  relations  between  the  true  and  apparent 
zenith  distances  and  azimuths.  All  the  quantities  in  the  second 
members  being  given,  the  first  two  equations  determine  J'sin  £', 
and  A ' ;  and  then  from  this  value  of  Jrsin  £',  and  that  of  */'cos  £ 
given  by  the  third  equation,  A'  and  £  are  determined. 


PARALLAX.  109 

But  it  is  convenient  to  introduce  the  horizontal  parallax 
instead  of  J.  For,  if  we  put  the  equatorial  radius  of  the  earth 
=  1,  we  have 

sin  T:  =  — 
J 

and  hence,  if  we  divide  the  equations  (97)  by  J,  and  put 


Has 


we  have 


/  sin  £'  cos  A'  =  sin  C  cos  A  — p  sin  r,  sin  (<p  —  p')       ^ 

/  sin  C'  sin  A'  =  sin  C  sin  A  I      (98) 

/  cos  C'  =  cos  C  —  />  sin  TT  cos  (p  —  9?')       j 

To  obtain  expressions  for  the  difference  between  £  and  f '  and 
between  J.  and  A',  that  is,  for  the  parallax  in  zenith  distance 
and  azimuth,  multiply  the  first  equation  of  (98)  by  sin  A,  the 
second  by  cos  A,  and  subtract  the  first  product  from  the  second ; 
again,  multiply  the  first  by  cos  J.,  the  second  by  sin  A,  and  add 
the  products:  we  find 
/  sin  C'  sin  (A'  —  A)  =  p  sin  r  sin  (p  —  ^')  sin  A 


/  sin  C'  cos  (A'  —  A)  —  sin  £  —  p  sin  r  sin  (<p  —  /)  cos  A       j 

Multiplying  the  first  of  these  by  sin  J  (A'  —  A),  the  second  by 
cos  J  (Af  — A),  and  adding  the  products,  we  find,  after  dividing 
the  sum  by  cos  J  (Af  — A\ 

<    cos  i  (A'  +  A) 
f  sin  C  =  sm  C  —  />  sm  TT  sin  (^  —  ^  ) 

which  with  the  third  equation  of  (98)  will  determine  £'. 
If  we  assume  f  such  that 

tan  Y  =  tan  (<p  —  <p')  -  (100) 

cos  \  (J/  —  A) 

we  have  the  following  equations  for  determining  £' : 

/  sin  C'  —  sin  C  —  />  sin  TC  cos  (^  —  ^')  tan  y        1 
/  cos  C'  =  cos  C  —  p  sin  r  cos  (y>  —  ^'')  /   / 

which,  by  the  process  employed  in  deducing  (99),  give 
/  sin  (C'  —  C)  =  p  sin  x  cos  (^  —  /) 


/  cos  (:'  —  C)  =  1  —  P  sin  TT  cos  (y  —  /)  cos  (:  ~ 

COS    * 


(102) 


110  PARALLAX. 

By  multiplying  the  first  of  these  by  sin  J  (£'  —  £),  the  second 
by  cos  J  (£'  —  £),  and  adding  the  products,  we  find,  after  dividing 
bycosi(C'-C), 


8n  *  cos     ~      cos 


cos  r  cos  i  (:'  —  c) 

or  multiplying  by  J, 

J'  =-  J  _  />  COS  (y  -  y')  COS  »  (C'  +  C)-  fi 

cos  r  cos  £  (£'  —  C) 

The  equations  (99)  determine  rigorously  the  parallax  in 
azimuth  ;  then  (100)  and  (102)  determine  the  parallax  in  zenith 
distance,  and  (103)  the  distance  of  the  star  from  the  observer. 

The  relation  between  J  and  J'  may  be  expressed  under  a  more 
simple  form.  By  multiplying  the  first  of  the  equations  (101)  by 
cos  ?-,  the  second  by  sin  f,  the  difference  of  the  products  gives 


sin  (C  —  Y) 

93.  The  preceding  formulae  may  be  developed  in  series. 

Put 

p  sin  TT  sin  (<p  —  ?>') 

sin  C 
then  (99)  become 

/  sin  C'  sin  (Af  —  A)  =  m  sin  C  sin  A 
f  sin  C*  cos  {A'  —  A)  =  sin  C  (1  —  m  cos  .4) 
whence 

tan  (A'  —  A)  =  — — (105) 

1  —  m  cos  A 

and  therefore  [PL  Trig.  Art.  258],  A'  —  A  being  in  seconds, 

jf^     —  ^^  __ •     •  i I™  *"i     0£C»  (  J.V/O ) 

Binl"          2sinl"         3sinl" 
To  develop  f  in  series,  we  take 

,N  cos  [-4  +  J  (A9  —  A)] 

tan  Y  =  tan  (^>  —  ?>') L- 

cos  £  (^4  —  ^4) 

=  tan  (<p  —  <pf)  [cos  A  —  sin  A  tan  $  ( J.'  —  J.)] 
whence,  by  interchanging  arcs  and  tangents  according  to  the 


PARALLAX.  Ill 

formulae  tan"1  y  =  y  —  J  if  +  &c.,  tan  x  =  x  +  i  £3  +  &c.  [PL 
Trig.  Arts.  209,  213], 

(<p  —  ?>')2  p  sin  TT  sin2  J.  sin  V 

Y  =  (<f>  —  y')  cos  A  —  ^-  -  -f  &c.     (107) 

-i  sin  C 


where  the  second  term  of  the  series  is  multiplied  by  sin 
because  /•  and  <p  —  <pf  are  supposed  to  be  expressed  in  seconds. 
Again,  if  we  put 


y>  sin  TT  cos  (<f> 


cos  f 
we  find  from  (102) 

tan(C'-t)  =  /Sin(C-'-\  (108) 

1  —  w  cos  (C  —  r) 

whence,  £'  —  f  being  in  seconds, 

nsin(:-r)       n'Bin2(C-r).«>Bin8(C-r)+Aa     (109) 
sin  1"  2  sin  1"  3  sin  1" 

Adding  the  squares  of  the  equations  (102),  we  have 

/2  =  f  —  jj  =  1  —  2  n  cos  (C  —  r)  +  ft2 
whence  [equations  (A)  and  (5),  Art.  82] 
log  J'  =  log  J  -  Jf    n  cos  (C  -  r)  +  n2  C°S  2(C  ~  ^  +  &c')    (11Q) 


wher^  Jf  =the  modulus  of  common  logarithms. 

94.  The  second  term  of  the  series  (107)  is  of  wholly  inappre- 
ciable effect  ;  so  that  we  may  consider  as  exact  the  formula 

Y  =  (<p  —  p')  cos  A  (111) 

and  the  rigorous  formulae  (105)  and  (108)  may  be  readily  com- 
puted under  the  following  form  : 

Put 

P  sin  TT  sin  (<p  —  of)  cos  A 
" 


then 


sin  *  =  m  cos  A  = 

sin  C 


1  —  sin 


A  =  tan  «  tan  (45°  +  J  *)  tan  A 


(112) 


112 

Put 

sin 
then 


PARALLAX. 


=  n  cos   C  -      = 


-  /)  cos  (C-  r) 


COS 


tan    C'  _      = 


1  —  sin  #' 
tan  #'  tan  (45°  -j-  i  #')  tan  (C  —  r) 


(113) 


EXAMPLE. — In  latitude  ^  =  38°  59;,  given  for  the  moon,  A  — 
320°  18',  £  =  29°  30',  and  n  =  58'  37".2,  to  find  the  parallax  in 
azimuth  and  zenith  distance. 

We  have  (Table  III.)  for  <p  =  38°  59',  <p  —  <p'  =  11'  15",  log  p 
-9.999428:  hence  by  (111)  r  =  8'39".3  and  £  —  r  =  29°  21' 
20".7 ;  with  which  we  proceed  by  (112)  and  (113)  as  follows : 


log  p  sin  TT 
log  sin  (0  — 
log  cosec  C 
log  cos  A 
18",  log  sin  & 
log  tan  •& 
log  tan  (45° 
log  tan  A 


8.23118 
7.51488 
0.30766 
9.88615 
6.93987 

5.93987 
0.00004 
n9.91919 
log  tan  (A1  —  A)  w5.85910 
A'  —  A  =  —  14".91 
A'  =  320°  17'  45".09 


log  p  sin  TT 

log  COS  (0  —  (j)') 

log  sec  y 
log  cos  (C  —  y) 
tf'=61'l".5,  log  sin  i?' 
log  tan  •&' 
log  tan  (45°  +  \ 
log  tan  (C  —  7) 
log  tan  (£'  —  C) 


=  29° 


8.231179 
9.999998 
0.000001 
9.940313 
8.171491 
8.171539 
0.006446 
9.750087 
7.928072 
29'  7".79 
59'  7".  79 


It  is  evident  that  we  may,  without  a  sacrifice  of  accuracy, 
omit  the  factors  cos  (tp  —  <p')  and  cos  f  in  the  computation  of  sin  $'. 

If  we  neglect  the  compression  of  the  earth  in  this  example, 
we  find  by  (94)  £'  —  C  =  29'  17".9,  which  is  10"  in  error.  * 

95.  To  find  the  parallax  of  a  star  in  zenith  distance  and  azimuth 
when  the  apparent  zenith  distance  and  azimuth  are  given,  the  earth 
being  regarded  as  a  spheroid. 

If  we  multiply  the  first  of  the  equations  (101)  by  cos  £'  and  the 
second  by  sin  £',  the  difference  of  the  products  gives 


sin  (C'  —  C)  = 


sin  TT  cos  (<f>  — 


sn    C  — 


cos 


for  which,  since  cos  (<p  —  ^')  and  cos  f  are  each  nearly  equal  to 
unity,  we  may  take,  without  sensible  error, 


sn 


C'  —  C)=p  sin  ?r  sin  (C'  — 


(114) 


PARALLAX.  113 

in  which  7-  has  the  value  found  by  (111),  or,  with  sufficient  accu- 
racy, by  the  formula 

r  =  (f  —  ?')  cos^L'  (115) 

Again,  if  we  multiply  the  first  of  the  equations  (98)  by  sin  Af 
and  the  second  by  cos  A1  ',  the  difference  of  the  products  gives 

')8in-4'  (116) 


sn 


to  compute  which,  £  must  first  be  found  by  subtracting  the  value 
of  the  parallax  £'  —  £,  found  by  (114),  from  the  given  value  of  £'. 

EXAMPLE.  —  In  latitude  <p  =  38°  59',  given  for  the  moon  A'  = 
320°  17'  45".09,  £'  =  29°  59'  7".79,  TT  =  58'  37".2,  to  find  the 
parallax  in  zenith  distance  and  azimuth. 

We  have,  as  in  the  example  Art.  94,  <p  —  <pf  =  11'  15",  log  p 

=  9.999428,  r  =  (<p  —  ?')  cos  A>  =  8'  39"-3>  C'  —  T  =  29°  50/  28"-5  ; 
and  hence,  by  (114)  and  (116), 

log  p  sin  TT  8.231179  log  p  sin  TT  8.23118 

log  sin  (C'  —  Y)      9.696879  log  sin  (?  —  <?')     7.51488 

log  sin  (C'  —  C)      7.928058  log  sin  A'  w9.80538 

C'  —  C  =  29'  7".79  log  cosec  :  0.30766 

C  =  29°  30'  0"  log  sin  (A'  —  A)  n5.85910 

A'  —  A  =  —  14".91 
^  =  320°  18'  0" 

agreeing  with  the  given  values  of  Art.  94. 

96.  For  the  planets  or  the  sun,  the  following  formulae  are  always 
sufficiently  precise  : 


A'  --  A  =  pir  sin  O  —  ?')  sin  A'  cosec  C' 


V 
j  ^ 


and  in  most  cases  we  may  take  £'  —  £  =  it  sin  £',  and  A'  —  A  =  0. 
The  quantity  px  is  frequently  called  the  reduced  parallax,  and 
it  —  />;:  —  (1  —  p)x  the  reduction  of  the  equatorial  parallax  for  the 
given  latitude  ;  and  a  table  for  this  reduction  is  given  in  some 
collections.  This  reduction  is,  indeed,  sensibly  the  same  as  the 
correction  given  in  our  Table  XIII.,  which  will  be  explained 
more  particularly  hereafter.  Calling  the  tabular  correction  Arc, 
we  shall  have,  with  sufficient  accuracy  for  most  purposes, 

pit  =  it  —  A:r 
VOL.  I.—  8 


114  PARALLAX. 

97.  The  preceding  methods  of  computing  the  parallax  enable 
us  to  pass  directly  from  the  geocentric  to  the  apparent  azimuth 
and  zenith  distance.  There  is,  however,  an  indirect  method 
which  is  sometimes  more  convenient.  This  consists  in  reducing 
both  the  geocentric  and  the  apparent  co-ordinates  to  the  point  in 
which  the  vertical  line  of  the  observer  intersects  the  axis  of  the  earth.  I 
shall  briefly  designate  this  point  as  the  point  0  (Fig.  11). 

We  may  suppose  the  point  0  to  be  assumed  as  the  centre  of 
the  celestial  sphere  and  at  the  same  time  as  the  centre  of  an 
imaginary  terrestrial  sphere  described  with  a  radius  equal  to  the 
normal  OA  (Fig.  11).  Since  the  point  0  is  in  the  vertical  line  of 
the  observer,  the  azimuth  at  this  point  is  the  same  as  the  appa- 
rent azimuth.  If,  therefore,  the  geocentric  co-ordinates  are  first 
reduced  to  the  point  0,  we  shall  then  avoid  the  parallax  in 
azimuth,  and  the  parallax  in  zenith  distance  will  be  found  by  the 
simple  formula  for  the  earth  regarded  as  a  sphere,  taking  the 
normal  as  radius. 

Since  the  point  0  is  in  the  axis  of  the  celestial  sphere,  the 
straight  line  drawn  from  it  to  the  star  lies  in  the  plane  of  the 
declination  circle  of  the  star;  the  place  of  the  star,  therefore,  as 
seen  from  the  point  0,  differs  from  its  geocentric  place  only  in 
declination,  and  not  in  right  ascension.  We  have  then  only  to 
find  the  reduction  of  the  declination  and  of  the  zenith  distance 
to  the  point  0. 

1st.  To  reduce  the  declination  to  the  point  0. — Let 
PP',  Fig.  13,  be  the  earth's  axis  ;  C  the  centre ; 
0  the  point  in  which  the  vertical  line  or  normal 
of  an  observer  in  the  given  latitude  <p  meets  the 
axis;  $  the  star.  We  have  found  for  CO  the 
expression  (Art.  84) 

CO  =  ai 

in  which  a  is  the  equatorial  radius  of  the  earth, 
and 

e*  sin  <p 
~  !/(!  —  <?  sin2  ? ) 

Let 

J  =  the  star's  geocentric  distance  SC, 

Jj  =  the  star's  distance  from  the  point  0       =  SO, 

d  •  =  the  geocentric  declination  —  90°  —  PCS, 

Sl  =  the  declination  reduced  to  the  point  0  =  90°  —  POS, 


PARALLAX.  115 

then,  drawing  SB  perpendicular  to  the  axis,  the  right  triangles 
SCB  and  SOB  give 


J4  sin  dt  =  A  sin  d  -j-  ai 
A.  cos  <5«  =  A  cos  d 


}     ("8) 
which  determine  4  and  ^.     From  these  we  deduce 


4  8in  (*i  —  *)  =  ««'  cos  *  \ 

J4  cos  (^  —  *)  ==  A  -f  <n  sin  5  J 

which  determine  Jt  and  the  reduction  of  the  declination.     If  we 
divide  these  by  J,  and  put 


in  which  n  denotes,  as  before,  the  equatorial  horizontal  parallax, 
they  become 

/j  sin  (di  —  <S)  =  e  sin  TT  cos  d 

ft  cos  («5t  —  <J)  =  1  -f  i  sin  ic  sin  S 
whence 

i  sin  K  cos  d 

tan  (dj  —  J)  = 

1  -j-  e  sin  TT  sin  <J 

or  in  series  [PL  Trig.  Art.  257], 

i  sin  it  cos  £       («  sin  7r)5  sin  2  £       . 

f~  sin  1"  2  sin  1" 

But  since  the  second  term  of  the  series  involves  i2  and  conse- 
quently e4,  and  this  is  further  multiplied  by  the  small  factor  sin2  TT, 
the  term  is  wholly  inappreciable  even  for  the  moon;  and,  as 
the  first  term  cannot  exceed  25"  in  any  case,  we  shall  obtain  ex- 
treme accuracy  by  the  simple  formula 

dl  —  d  ==  i  *  cos  <5  (120) 

The  value  of  4  is  found  from  (119),  by  the  same  process  as 
was  used  in  finding  Jr  in  (103),  to  be 

(  -,  fiin  J  (8,  4-  £)  ) 

A.  —  A  <  1  -K.I  sin  TT " — - — -  \ 

\  cos  \  (dt  —  d)  ) 

or,  on  account  of  the  small  difference  between  dl  and  d, 

At  =  A  (1  -f  i  sin  TT  sin  8)  (121) 


116 


PARALLAX. 


As  ^  -  d  is  so  small,  it  may  be  accurately  computed  with 
logarithms  of  four  decimal  places,  and  it  will  be  convenient  to 
substitute  for  i  the  form 


in  which 


i  =  A  sin  <p 


A  = 


y\\  —  e2  sin2  p) 

The  value  of  log  A  may  then  be  taken  from  the  following 
table  with  the  argument  <p  =  the  geographical  latitude 


* 

log  A 

0° 

7.8244 

10 

7.8245 

20 

7.8246 

30 

7.8248 

40 

7.8250 

50 

7.8253 

60 

7.8255 

70 

7.8257 

80 

7.8258 

90 

7.8259 

We  shall  then  compute  ^  --  d  and  Jx  under  the  following 
forms : 

^  —  d  —  A  1:  sin  <p  cos  d 

J4  =  J  (1  -f-  A  sin  TT  sin  <p  sin  <5) 


}    (122) 


If  the  value  of  ^  has  been  found  as  below,  we  may  take 
dt  —  d  =  e2  TTj  sin  <p  cos  d 

2d.  To  find  the  parallax  in  zenith  distance  for  the  point  0. — Let 
ZAO,  Fig.  14,  be  the  vertical  line  of  the  observer  at 

^Fig.  14.  ^  The  normal  J.O  terminating  in  the  axis  being 
denoted  by  N9  we  have,  by  (90), 


—  e  sn    y 

But  if  in  (84)  we  write  e4  sin4  <p  for  e4  sin2  ^,  we  have 
p  =  a  |/(1  —  e2  sin2  <p) 

and  this  value  is  sufficiently  accurate  for  the  compu- 
tation of  the  parallax  in  all  cases.  If  then  we  put 
a  =  1,  we  have 


PARALLAX.  117 

_  J. 

~   P 

If  now  in  the  vertical  plane  passing  through  the  line  ZO  and 
the  star  8  we  draw  SB  perpendicular  to  OZ,  and  put 

C4  =  the  zenith  distance  at  O  =  SOZ 
C'  =  the  apparent  zenith  dist.  =  SAZ 

the  triangles  OSB,  ASB  give 

J'cosC'^cosd-1  |    (m) 

J'  sin  C'  =  AI  sin  Ct  J 

Dividing  these  equations  by  4,  and  putting 

—  =fl        sin  ^  = 

they  become 

ft  cos  C'  =  cos  d  —  sin  r1 
/4  sin  C'  =  sin  C4 

from  which  we  deduce 

/t  sin  (C'  —  Ct)  =  sin  ^  sin  C4 

/t  cos  (C'  —  CJ  =  1  —  sin  rt  cos  Ct 


tan  (C'  -  C,)  =  ,    8n.'r.sn.  (124) 

1  —  sin       cos  C 


and  in  series, 


Or,  rigorously, 

sin  ft  =  sin  ^  cos  C4  | 

tan  (:'  —  ct)  =  tan  «  tan  (45°  +  J  *)  tan  Ct  j 

To  find  ^  we  have 

1  1 

sin  TT,  =  -  = 


1        /o Jj         /o  J  (1  -j-  4  sin  rr  sin  ?>  sin  ^) 

or  gin  »,  = ^^ .  (127) 

/o(l  -f-  A  sin  r  sin  <p  sin  ^) 


118  PARALLAX. 

But  this  very  precise  expression  of  ^  will  seldom  be  required : 
it  will  generally  suffice  to  take 


sin 

sm  TT,  = 


or  JT.  =  — 


P 


which  will  be  found  to  give  the  correct  value  of  icl9  even  for  the 
moon,  within  0".2  in  every  case.  Where  this  degree  of  accu- 
racy suffices,  we  may  employ  a  table  containing  the  correction 
for  reducing  jr  to  itv  computed  by  the  formula 


Table  XIII.,  Vol.  II.,  gives  this  correction  with  the  arguments  n 
and  the  geographical  latitude  (p.  Taking  the  correction  from 
this  table,  therefore,  we  have 

7T4  =  TT  -f  A*  (128) 

Sd.  To  compute  the  parallax  in  zenith  distance  for  the  point  0  ivhen 
the  apparent  zenith  distance  is  given. 

Multiplying  the  lirst  equation  of  (123)  by  sin  f',  the  second  by 
cos  £',  and  subtracting,  we  find 

sin  (:'  —  CJ  — sin  £' 

M 
or  sin  (;'  —  C4)  —  sin  r1  sin  C'  (129) 

If  we  denote  the  apparent  altitude  by  hf  and  the  altitude 
reduced  to  the  point  0  by  h^  this  equation  becomes 

sin  (\  —  A')  =  sin  r,  cos  hf  (130) 

EXAMPLE. — In  Latitude  <p  =  38°  59',  given  the  moon's  hour 
angle  t  =  341°  V  36".85,  geocentric  declination  3  =  -f  14°  39' 
24".54,  and  the  equatorial  horizontal  parallax  TT  =  58r  37".2,  to 
find  the  apparent  zenith  distance  and  azimuth. 

The  geocentric  zenith  distance  and  azimuth,  computed  from 
these  data  by  Art.  14,  are  f  =  29°  30',  A  =  320°  18',  which  are 
the  values  employed  in  our  example  in  Art.  94.  To  compute 


PARALLAX.  119 

by  the  method  of  the  present  article,  we  first  reduce  the  declina- 
tion to  the  point  0  by  (122),  as  follows : 

For  <f>  =  38°  59'        log  A  7.8250 

TT  =  3517".2        log  *  3.5462 

log  sin  <p     9.7987 

8  =  14°  39'  24".54        log  cos  3     9.9856 
dt—  8=  14.31         log  (<5,  — <5)  1.1555 

(5±:=:14039'38".85 

With  this  value  of  ^  and  /  =  341°  1'  a6".85,  the  computation 
of  the  zenith  distance  and  azimuth  by  Art.  14  gives  for  the 
point  0 

Ct  =  29°  29'  47".67  At  =  320°  17'  45".09 

and  this  value  of  A l  is  precisely  the  same  as  A'  found  in  Art.  94, 
as  it  should  be,  since  the  azimuth  at  the  point  0  and  at  the 
observer  are  identical. 

We  find  from  Table  XIII.  ATT  =  4".6,  and* hence nl=  58'  37".2 
+  4".6  =  58'  41".8;  and  then,  by  (126), 

log  sin  7rt  8.23232 

log  cos  Ct  9.93971 

#  ==  51'  5"  log  sin  0  8.17203 

log  tan  #  8.17208 

log  tan  (45°  +  i  *)  0.00645 

Ct  =  29°  29'  47".67  log  tan  Ct  9.75258 

C'  —  ct  =        29  20  .03  log  tan  (;'  —  CJ  7.93111 

C'  =  29°  59'    7".70 

agreeing  with  the  value  found  in  Art.  94  within  0".09.  If  we 
had  computed  jfj  by  (127),  the  agreement  would  have  been  exact. 

98.  To  find  the  parallax  of  a  star  in  right  ascension  and  declination 
when  its  geocentric  right  ascension  and  declination  are  given. 

The  investigation  of  this  problem  is  similar  to  that  of  Art.  92. 
Let  the  star  be  referred  by  rectangular  co-ordinates  to  three 
planes  passing  through  the  centre  of  the  earth :  the  first,  the 
plane  of  the  equator ;  the  second,  that  of  the  equinoctial  colure  ; 
the  third,  that  of  the  solstitial  colure.  Let  the  axis  of  x  be  the 
straight  line  drawn  through  the  equinoctial  points,  positive 
towards  the  vernal  equinox ;  the  axis  of  i/,  the  intersection  of 


120  PARALLAX. 

the  plane  of  the  solstitial  colure  and  that  of  the  equator,  positive 
towards  that  point  of  the  equator  whose  right  ascension  is  90°  ; 
the  axis  of  z,  the  axis  of  the  heavens,  positive  towards  the  north. 

Let 

a  =  the  star's  geocentric  right  ascension, 

5  =  "  "         declination, 
J  ==             "               "          distance, 

then  the  co-ordinates  of  the  star  are 

x  =  A  cos  S  cos  a 
y  =  J  cos  <5  sin  a 

z  =  J  sin  d 

Again,  let  the  star  be  referred  to  another  system  of  planes 
parallel  to  the  first,  the  origin  being  the  observer.  The  vanish- 
ing circles  of  these  planes  in  the  celestial  sphere  are  still  the 
equator,  the  equinoctial  colure,  and  the  solstitial  colure.  Let 

a'  =  the  star's  observed  right  ascension, 

6  =  "  "         declination, 

J'  =  "      distance  from  the  observer, 

where  by  observed  right  ascension  and  declination  we  now  mean 
the  values  which  differ  from  the  geocentric  values  by  the  paral- 
lax depending  on  the  position  of  the  observer  on  the  surface  of 
the  earth.  The  co-ordinates  of  the  star  in  this  system  will  be 

x'  =  J'  cos  <$'  cos  a! 
y'  =  J'  cos  d'  sin  a' 
z'  =  J'  sin  (T 
Now,  if 

0  =  the  sidereal  time  =  the  right  ascension  of  the  observer's 

meridian  at  the  instant  of  observation, 
<p<  =  the  reduced  latitude  of  the  place  of  observation, 
p  =  the  radius  of  the  earth  for  this  latitude, 

then  0,  /,  and  p  are  the  polar  co-ordinates  of  the  observer, 
entirely  analogous  to  a,  <5,  and  J  of  the  star,  so  that  the  rectan- 
gular co-ordinates  of  the  observer,  taken  in  the  first  system,  are 

a  =  p  cos  y'  cos  0 
b  =  p  cos  <pr  sin  0 
c  =  p  sin  <f 


PARALLAX.  121 

arid  for  transformation  from  one  system  to  the  other  we  have 

x'  =  x  —  a,         y'  =  y  —  6,         zf  =  z  —  c. 


Hence 

J'  COS  3'  COS  a'  =  A  COS  3  COS  a  —  p  COS  <f>'  COS  0 

A'  cos  3'  sin  a'  =  A  cos  <5  sin  a  —  />  cos  ?>'  sin  0 
J'  8in  5'  =  J  sin  <S  -  />  sin  <?' 

or,  dividing  by  J,  and  putting  as  before 

J'  1 


^  


(131) 


Bin  rr  ==. 


/  COS  <5'  COS  a'  =  COS  3  COS  a  —  /o  Sin  ic  COS  /  COS  0 

/  cos  3'  sin  a!  =  cos  <5  sin  a  —  p  sin  :r  cos  ?>'  sin  0 
/  sin  3'  =  sin  3  —  p  sin  n  sin  ^' 

From  the  first  two  of  these  equations  we  deduce 


(132) 


/  cos  3'  sin  (a'  —  a)  =  p  sin  it  cos  <?'  sin  (a  —  0)  ) 

/  COS  ff  COS  (a'  —  a)  =  COS  3  —  p  sin  7T  COS  <p'  COS  (a  —   0)    J       ^         * 

Multiplying  the  first  of  these  by  sin  J  (a'  —  a),  the  second  by 
cos  f  (a'  —  a),  and  adding  the  products,  we  find,  after  dividing  by 

cos  \  (a'  —  a), 


Put 


/cos  *=  cos  3  - 


tan 


COS  £  (a'  —  a) 
tan  <p'  cos  £  (a'  —  a) 

COS  [J  "(a'  -f  a)  —  0] 


then  we  have,  for  determining  £', 


/  sin  3'  =  sin  3  —  p  sin  it  sin  ^/ 
/  cos  3f  =  cos  5  —  p  sin  TT  sin  ?>'  cot 


whence 


,..,., 
f  sin  (3f  —  5)  =  p  sin  TT  sin  <P 


.  sin  (3  — 


sin 


.  COS  (5  —  y) 

f  coa  (3*  —  3)  =  1  —  p  sin  w  sin  /  -  V-- 

sin 


.  _   A'         sin  (^  — 

''  ~~     ''  =       ~' 


(134) 


}    (135) 


(136) 


(137) 


The  equations  (133)  determine,  rigorously,  the  parallax  in  right 


122  PARALLAX. 

ascension, or  a'  —  a ;  (136)  the  parallax  in  decimation,  or  3f  —  d; 
and  (137)  determines  J'. 

99.  To  obtain  the  developments  in  series,  put 

p  sin  TT  cos  <p' 


m  = 


cos  d 


then  from  (133)  we  have 

m  sin  (a  —  0") 

tan  (a'  —  a)  =  =—  — 4rv  (138) 

1  —  m  cos  (a  —  0) 

whence 

'  _         -  m  8in  (°  ~  ®)    i    m*  8in  2  (a  —  e)     ,     <L 
sin  1"  2  sin  1" 

Putting 

P  sin  TT  sin  «>' 


sin 
we  have  from  (136) 


n  sin  (5  —  v) 

tan  (^  —  5)  =  -—  n  40) 

1  —  n  cos    5  —  y 


whence 


4 100.  The  quantity  a  —  0  is  the  hour  angle  of  the  star  east  of 
the  meridian.  According  to  the  usual  practice,  we  shall  reckon 
the  hour  angle  towards  the  west,  and  denote  it  by  /,  or  put 

t=  0  —a 

and  then  we  shall  write  (138)  and  (140)  as  follows : 

* 

m  sin  t 
tan  (a  —  a')  = 


1  —  m  cos  t 


1 

1  —  n  cos       — 


The  rigorous  computation  will  be  conveniently  performed  by 
the  following  formulae  : 


sin  #  =  m  cos  t  — 


PARALLAX.  123 

p  sin  r  cos  ^'  cos  t 


cos  5 
tan  (a  —  a')  =  tan  #  tan  (45°  -f  M)  tan  # 

tan  <f>'  cos  J  (o  —  a') 
tan  Y  —  — 

COS  [*  -MO-"')] 

/>  sin  TT  sin  «>'  cos  (y  — 
sm  &'  =  n  cos  (>  —  <5)  =  " 


sin  f 

tan  ((5  —  (T)  =  tan  #'  tan  (45°  +  i  #')  tan  0  —  a) 

101.  Except  for  the  moon,  the  first  terms  of  the  series  (139) 
and  (141)  will  suffice,  and  we  may  use  the  following  approxi- 
mations : 


pi:  cos  <?'  sin  t 


cos  3 


tan     = 


tan 


cos  t 


sn 


(143) 


If  the  star  is  on  the   meridian,  we  have  t  =  0,  and  hence 

f  =  <f>',  and 

d  —  <5'  =  pn  sin  (y  —  5) 

Since  in  the  meridian  we  have  £  =  <p  —  <5,  it  is  easily  seen 
that  £'  —  C  found  by  (108)  and  tf'  —  ^.found  by  (140)  will  then 
be  numerically  equal,  or  the  parallax  in  zenith  distance  is  numeri- 
cally equal  to  the  parallax  in  declination  when  the  star  is  on  the  meri- 
dian. 

102.  To  find  the  parallax  of  a  star  in  right  ascension  and  declination, 
when  its  observed  right  ascension  and  declination  are  given. 

Multiplying  the  first  equation  of  (132)  by  sin  a',  the  second 
by  cos  ar,  and  subtracting  one  product  from  the  other,  we  find 


Sin  (a  -  a')  = 


cos  3 
In  like  manner,  from  (135)  we  deduce 


124  PARALLAX. 


sn 


We  have  here  0  —  a'  equal  to  the  apparent  or  observed  hour 
angle  ;  and  hence,  putting 

t'  =  0  —  a' 
the  computation  may  be  made  under  the  following  form  : 

p  sin  TT  cos  <p'  sin  tf 


Sin  (a  —  a')  = 

tan     = 


cos  3 
tan  <f>'  cos  $  (a  —  a') 


COS  p'  —  J  (a  —  a')] 

p  sin  TT  sin  <p'  sin  O  — 
sin  (<5  —  <5')  =  — 


sn 


(144) 


In  the  first  computation  of  a  —  a'  we  employ  df  for  8.  The 
value  of  a  —  a'  thus  found  is  sufficiently  exact  for  the  compu- 
tation of  f  and  8  —  8f.  With  the  computed  value  of  8  —  8f  we 
then  find  8  and  correct  the  computation  of  a  —  a'. 

EXAMPLE. — Suppose  that  on  a  certain  day  at  the  Greenwich 
Observatory  the  right  ascension  and  declination  of  the  moon 
were  observed  to  be 

a'  =  7*  41-  2CK436 
if  =  15°  50'  27".66 

when  the  sidereal  time  was 

0  =  11*  I7m  (K02 
and  the  moon's  equatorial  horizontal  parallax  was 

TT  =  56'  57".5 
Required  the  geocentric  right  ascension  and  declination. 

We  have  for  Greenwich  <p  =  51°  28'  38".2,  and  hence  (Table  III.') 
?  —  j  =  11'  13".6,  /  =  51°  17'  24".6,  log  p  =  9.9991134.  The  com- 
putation by  (144)  is  then  as  follows : 


PARALLAX. 


125 


a'  (in 

e 
*(•- 

log  sec  \t'  — 
log  cos  £  (a  - 
log  tan  $' 
log  tan  7 

log  p  sin  TT 
log  sin  0' 
log  sin  (7  — 
log  cosec  7 
log  sin  (<J  — 
d 

arc)  =  115°  20*    6".54 
«       =  169    15     0  .30 

log  /)  sin  TT 
log  cos  ^' 
log  sin  t' 

(1) 
log  cos  6' 

App.  log  sin  (a  — 
Approx.  a  —  a'  ; 

(1)         - 
log  cos  6 

log  sin  (a  —  a') 
a  —  a'  = 

a  = 

8.218377 
9.796142 
9.907489 

«'  =    53    54  53  .76 

-  a')  =           14  55  .8 
-  a')  =     53    39  58 

J(a  —  a')]     0.227319 
-  a')                 9.999996 
0.096133 

7.922008 
9.983185 

a')      7.938823 
=        29'  51".6 
7.922008 
9.981835 

0.323448 

7  =    64°  35'  58" 
—  6'=    48    45  30 

8.218377 
9.892275 
d')                     9.876181 
0.044153 

7.940173 

4-    29'57".23 

115°  50'  3".77 
7*  43"»  20*.  251 

<J')                     8.030986 

—  6'=     4-  36'55".24 
<J  =  16°  27'  22".90 

103.  For  all  bodies  except  the  moon,  the  second  computation 
will  never  affect  the  result  in  a  sensible  degree,  and  we  may  use 
the  following  approximations : 


sin  f 


3  —  #  = 


sin  Y 


(145) 


For  the  sun,  planets,  and  comets,  it  is  frequently  more  conve- 
nient to  use  the  geocentric  distance  of  the  body  instead  of  the 
parallax,  or,  at  least,  to  deduce  the  parallax  from  the  distance, 
the  latter  being  given.  This  distance  is  always  expressed  in 
parts  of  the  sun's  mean  distance  as  unity.  If  we  put 

TTO  =  the  sun's  mean  equatorial  horizontal  parallax, 
J0  =  the  sun's  mean  distance  from  the  earth, 

we  have,  whatever  unit  is  employed  in  expressing  J,  J0,  and  a, 


a 

sm  TT  =  - 
J 


sin  rn  =  — 


126  PARALLAX. 

whence 

sin  TT  =  —  sin  ?r( 
A 

and  when  we  take  J0  —  1, 


(146) 


According  to  ENCKE'S  determination 

*Q=  8".57116        log  TTO  =  0.93304 

EXAMPLE.  —  DONATI'S  comet  was  observed  by  Mr.  JAMES  FER- 
GUSON at  Washington,  1858  Oct.  13,  6'1  26m  21M  mean  time, 
and  its  observed  right  ascension  and  declination  when  corrected 
for  refraction  were 

a'  =  236°  48f    0".5 
S'  =  —  T°  36'  52".8 

The  logarithm  of  the  comet's  distance  from  the  earth  was  log  A 
—  9.7444.     Required  the  geocentric  place. 

We  have  for  Washington  <p  =  38°  53'  39".3,  whence,  by  Table 
m.,  log  p  cos  <pf  —  9.8917,  log  p  sin  <p'  =  9.7955.  Converting  the 
mean  into  sidereal  time  (Art.  50),  we  find  0  =  19*  55TO  16*.98. 
Hence,  by  (145)  and  (146), 

0  =  298°  49'.2  log  tan  ?'  9.9038 

a'  =  236    48.0  log  cos  if  9.6713 

t'  =    62     1  .2  log  tan  r  0^2325" 

log  TTO  0.9330  r          =  59°  39'.2 

log  J  9.7444  r  —  if  =  67    16  .1 

logr  1.1886 

log  pit  cos  y'    1.0803  log  pTt  sin  <p*  0.9841 

log  sin  t'         9.9460  log  sin  (r  —  *)  ,9.9649 

log  sec  «5'        0.0038  log  eosec  >-  0.0640 

log  (a  —  a')  T6301  log  (3  —  #)  1.0130 

a  —  a'  .=  +  10".7  *  —  if  =  +  10'f.3 

Hence,  for  the  geocentric  place  of  the  comet, 

.  a  =  236°  48'  11".2  9  =  —  7°  36'  42".5 

104.  Parallax  in  latitude  and  longitude.  —  Formulae  similar  to  the 
ubove  obtain  for  the  parallax  in  latitude  and  longitude.  We 


REFRACTION.  127 

• 

have  only  to  substitute  for  0  and  <pf  (which  are  the  right  ascension 
and  declination  of  the  geocentric  zenith)  the  corresponding 
longitude  and  latitude  of  the  geocentric  zenith  (which  will  be 
found  by  Art.  23),  and  put  X  and  /?  in  the  place  of  a  and  3.  Thus, 
if  I  and  b  are  the  longitude  and  latitude  of  the  geocentric  zenith, 
the  equations  (143)  give  for  all  objects  except  the  moon. 


,  _  pi:  cos  6  sin  (I  — 

A    -   A      —  — 


cos 


tan     = 


tan  b 


cos  (I  —  /) 


sn 


(147) 


In  the  game  manner,  the  equations  (131)  may  be  made  to 
express  the  general  relations  between  the  geocentric  and  the 
apparent  longitude  and  latitude,  and  for  the  moon  we  can 
employ  (142),  observing  to  substitute  respectively 

for  a,       a',        3,         d',         0,         <p' 
the  quantities  >*,        J',         £,         /§',         I,  b 

In  all  the  formulae,  when  we  choose  to  neglect  the  compression 
of  the  earth,  we  have  only  to  put  <p  =  <p'  and  p  =  l. 

REFRACTION. 

105.  General  laws  of  refraction. — The  path  of  a  ray  of  light  is  a 
straight  line  so  long  as  the  ray  is  passing  through  a  medium  of 
uniform  density,  or  through  a  vacuum.  But  when  a  ray  passes 
obliquely  from  one  medium  into  another  of  different  density,  it 
is  bent  or  refracted.  The  ray  before  it  enters  the  second  medium 
is  called  the  incident  ray ;  after  it  enters  the  second  medium  it  is 
called  the  refracted  ray;  and  the  difference  between  the  directions 
of  the  incident  and  refracted  rays  is  called  the  refraction. 

If  a  normal  is  drawn  to  the  surface  of  the  refracting  medium 
at  the  point  where  the  incident  ray  meets  it,  the  angle  which  the 
incident  ray  makes  with  this  normal  is  called  the  angle  of  inci- 
dence, the  angle  which  the  refracted  ray  makes  with  the  normal 
is  the  angle  of  refraction,  and  the  refraction  is  the  difference  of 
these  two  angles. 


128  REFRACTION. 

Thus,  if  SA,  Fig.  15,  is  an  incident  ray  upon  the  surface  BB ' 
of  a  refracting  medium,  AC  the  refracted 
ray,  MN  the  normal  to  the  surface  at  A, 
SAM  is  the  angle  of  incidence,  CAN  is  the 
angle  of  refraction  ;  and  if  CA  he  produced 
backwards  in  the  direction  AS',  SASf  is  the 
refraction.  An  observer  whose  eye  is  at 
any  point  of  the  line  AC  will  receive  the 
ray  as  if  it  had  come  directly  to  his  eye 
without  refraction  in  the  direction  S'AC, 
which  is  therefore  called  the  apparent 
direction  of  the  ray. 

Now,  it  is  shown  in  Optics  that  this  refraction  takes  place 
according  to  the  following  general  laws : 

1st.  When  a  ray  of  light  falls  upon  a  surface  (of  any  form) 
which  separates  two  media  of  different  densities,  the  plane  which 
contains  the  incident  ray  and  the  normal  drawn  to  the  surface 
at  the  point  of  incidence  contains  the  refracted  ray  also. 

2d.  When  the  ray  passes  from  a  rarer  to  a  denser  medium,  it 
is  in  general  refracted  towards  the  normal,  so  that  the  angle  of 
refraction  is  less  than  the  angle  of  incidence ;  and  when  the  ray 
passes  from  a  denser  to  a  rarer  medium,  it  is  refracted  from  the 
normal,  so  that  the  angle  of  refraction  is  greater  than  the  angle 
of  incidence. 

3d.  Whatever  may  be  the  angle  of  incidence,  the  sine  of  this 
angle  bears  a  constant  ratio  to  the  sine  of  the  corresponding 
angle  of  refraction,  so  long  as  the  densities  of  the  two  media  are 
constant.  If  a  ray  passes  out  of  a  vacuum  into  a  given  medium, 
the  number  expressing  this  constant  ratio  is  called  the  index  of 
refraction  for  that  medium.  This  index  is  always  an  improper 
fraction,  being  equal  to  the  sine  of  the  angle  of  incidence  divided 
by  the  sine  of  the  angle  of  refraction. 

4th.  When  the  ray  passes  from  one  medium  into  another,  the 
sines  of  the  angles  of  incidence  and  refraction  are  reciprocally 
proportional  to  the  indices  of  refraction  of  the  two  media. 

106.  Astronomical  refraction. — The  rays  of  light  from  a  star  in 
coming  to  the  observer  must  pass  through  the  atmosphere  which 
surrounds  the  earth.  If  the  space  between  the  star  and  the 
upper  limit  of  the  atmosphere  be  regarded  as  a  vacuum,  or  as 
filled  with  a  medium  which  exerts  no  sensible  effect  upon  the 


REFRACTION.  129 

direction  of  a  ray  of  light,  the  path  of  the  ray  will  be  at  first  a 
straight  line;  but  upon  entering" the  atmosphere  its  direction 
will  be  changed.  According  to  the  second  law  above  stated,  the 
new  medium  being  the  denser,  the  ray  will  be  bent  towards  the 
normal,  which  in  this  case  is  a  line  drawn  from  the  centre  of  the 
earth  to  the  surface  of  the  atmosphere  at  the  point  of  incidence. 

The  atmosphere,  however,  is  not  of  uniform  density,  but  is 
most  dense  near  the  surface  of  the  earth,  and  gradually  decreases 
in  density  to  its  upper  limit,  where  it  is  supposed  to  be  of  such 
extreme  tenuity  that  its  first  effect  upon  a  ray  of  light  may  be 
considered  as  infinitesimal.  The  ray  is  therefore  continually  pass- 
ing from  a  rarer  into  a  denser  medium,  and  hence  its  direction 
is  continually  changed,  so  that  its  path  becomes  a  curve  which 
is  concave  towards  the  earth. 

The  last  direction  of  the  ray,  or  that  which  it  has  when  it 
reaches  the  eye,  is  that  of  a  tangent  to  its  curved  path  at  this 
point;  and  the  difference  of  the  direction  of  the  ray  before  en- 
tering the  atmosphere  and  this  last  direction  is  called  the  astro- 
nomical refraction,  or  simply  the  refraction. 

Thus,  Fig.  16,  the  ray  Se  from  a  star,  entering  the  atmosphere 
at  e,  is  bent  into  the  curve  ecA 
which  reaches  the  observer  at  A  in  ^ 
the  direction  of  the  tangent  S'A 
drawn  to  the  curve  at  A.  If  CAZ 
is  the  vertical  line  of  the  observer, 
or  normal  at  J.,  by  the  first  law  of 
the  preceding  article,  the  vertical 
plane  of  the  observer  which  con- 
tains the  tangent  AS'  must  also 
contain  the  whole  curve  Ae  and 
the  incident  ray  Se.  Hence  refrac- 
tion increases  the  apparent  altitude 
of  a  star,  but  does  not  affect  its  azi- 
muth. 

The  angle  SfAZ  is  the  apparent  ze- 
nith distance  of  the  star.     The  true  zenith  distance*  is  strictly  the 
angle  which  a  straight  line  drawn  from  the  star  to  the  point  A 

*  By  true  zenith  distance  we  here  (and  so  long  as  we  are  considering  only  the 
effect  of  refraction)  mean  that  which  differs  from  the  apparent  zenith  distance  only 
by  the  refraction. 
VOL.  L— 9 


130  REFRACTION. 

makes  with  the  vertical  line.  Such  a  line  would  not  coincide 
with  the  ray  Se;  but  in  consequence  of  the  small  amount  of  the 
refraction,  if  the  line  Se  be  produced  it  will  meet  the  vertical 
line  AZ  at  a  point  so  little  elevated  above  A  that  the  angle 
which  this  produced  line  will  make  with  the  vertical  will  differ 
very  little  from  the  true  zenith  distance.  Thus,  if  the  produced 
line  Se  be  supposed  to  meet  the  vertical  in  6',  the  difference 
between  the  zenith  distances  measured  at  b'  and  at  A  is  the 
parallax  of  the  star  for  the  height  Ab',  and  this  difference  can  be 
appreciable  only  in  the  case  of  the  moon.  It  is  therefore  usual 
to  assume  Se  as  identical  with  the  ray  that  would  come  to  the 
observer  directly  from  the  star  if  there  were  no  atmosphere. 

The  only  case  in  which  the  error  of  this  assumption  is  appre- 
ciable will  be  considered  in  the  Chapter  on  Eclipses. 

107.  Tables  of  .Refraction . — For  the  convenience  of  the  reader 
who  may  wish  to  avail  himself  of  the  refraction  tables  without 
regard  to  the  theory  by  which  they  are  computed,  I  shall  first 
explain  the  arrangement  and  use  of  those  which  are  given  at 
the  end  of  this  work. 

Since  the  amount  of  the  refraction  depends  upon  the  density 
of  the  atmosphere,  and  this  density  varies  with  the  pressure  and 
the  temperature,  which  are  indicated  by  the  barometer  and  the 
thermometer,  the  tables  give  the  refraction  for  a  mean  state  of 
the  atmosphere;  and  when  the  true  refraction  is  required,  supple- 
mentary tables  are  employed  which  give  the  correction  of  the 
mean  refraction  depending  upon  the  observed  height  of  the 
barometer  and  thermometer. 

TABLE  I.  gives  the  refraction  when  the  barometer  stands  at 
30  inches  and  the  thermometer  (Fahrenheit's)  at  50°.  If  we 
put 

r  —  the  refraction, 

z  =  the  apparent  zenith  distance, 

£  =  the  true  zenith  distance, 
then 

:  =  z  +  r 

Where  great  accuracy  is  not  required,  it  suffices  to  take  r 
directly  from  TABLE  I.  and  to  add  it  to  z.  (The  resulting  £  is 
that  zenith  distance  which  we  have  heretofore  denoted  by  £'  in 
the  discussion  of  parallax.)  The  argument  of  this  table  is  the 
apparent  zenith  distance  z. 


REFRACTION.  131 

TABLE  II.  is  BESSEL'S  Refraction  Table,*  which  is  generally 
regarded  as  the  most  reliable  of  all  the  tables  heretofore  con- 
structed. In  Column  A  of  this  table  the  refraction  is  regarded 
as  a  function  of  the  apparent  zenith  distance  z,  and  the  adopted 
form  of  this  function  is 

r  =  o.pAY*  tan  z 

in  which  a  varies  slowly  with  the  zenith  distance,  and  its  loga- 
rithm is  therefore  readily  taken  from  the  table  with  the  argu- 
ment z.  The  exponents  A  and  ^  differ  sensibly  from  unity  only 
for  great  zenith  distances,  and  also  vary  slowly;  their  values  are 
therefore  readily  found  from  the  table. 

The  factor  ft  depends  upon  the  barometer.  The  actual  pres- 
sure indicated  by  the  barometer  depends  not  only  upon  the 
height  of  the  column,  but  also  upon  its  temperature.  It  is, 
therefore,  put  under  the  form 

0  =  BT 

and  log  B  and  log  77are  given  in  the  supplementary  tables  with 
the  arguments  "height  of  the  barometer,"  and  "height  of  the 
attached  thermometer,"  respectively  ;  so  that  we  have 

log  ft  =  log  B  -f  log  T 

Finally,  log  f  is  given  directly  in  the  supplementary  table  with 
the  argument  "  external  thermometer."  This  thermometer  must 
be  so  exposed  as  to  indicate  truly  the  temperature  of  the  atmo- 
sphere at  the  place  of  observation. 

In  Column  B  of  the  table  the  refraction  is  regarded  as  a 
function  of  the  true  zenith  distance  £  expressed  under  the  form 

r  =  a'/9^'r*'tan  C 

and  log  a',  Af,  and  Xf  are  given  in  the  table  with  the  argument  £ ; 
ft  and  f  being  found  as  before. 

Column  A  will  be  used  when  z  is  given  to  find  £ ;  and  Column 
B,  when  £  is  given  to  find  z. 

Column  C  is  intended  for  the  computation  of  differential  re- 
fraction, or  the  difference  of  refraction  corresponding  to  small 


*  From  his  Astronomisehe  Untersuchungen,  Vol.  I. 


132 


REFRACTION. 


differences  of  zenith  distance,  and  will  be  explained  hereafter 
(Miorometric  Observations,  Vol.  II.). 

These  tables  extend  only  to  85°  of  zenith  distance,  beyond 
which  no  refraction  table  can  be  relied  upon.  There  occur  at 
times  anomalous  deviations  of  the  refraction  from  the  tabular 
value  at  all  zenith  distances;  and  these  are  most  sensible  at 
great  zenith  distances.  Fortunately,  almost  all  valuable  astrono- 
mical observations  can  be  made  at  zenith  distances  less  than 
85°,  and  indeed  less  than  80°  ;  and  within  this  last  limit  we 
are  justified  by  experience  in  placing  the  greatest  reliance  in 
BESSEL'S  Table.  In  an  extreme  case,  where  an  observation  is 
made  within  5°  of  the  horizon,  we  can  compute  an  approximate 
value  of  the  refraction  by  the  aid  of  the  following  supplement- 
ary table,  which  is  based  upon  actual  observations  made  by 
ARGELANDER.* 


App.  zen. 
distance. 

log  Refract. 

A 

A 

85°  0' 

2.76687 

1.0127 

1.1229 

30 

2.80590 

1.0147 

1.1408 

86  0 

2.84444 

1.0172 

1.1624 

30 

2.88555 

1.0204 

1.1888 

87  0 

2.93174 

1.0244 

1.2215 

80 

2.98269 

1.0298 

1.2624 

88  0 

3.03686 

1.0368 

1.3141 

30 

3.09723 

1.0465 

1.3797 

89  0 

3.16572 

1.0593 

1.4653 

30 

3.24142 

1.0780 

1.5789 

If  we  call  R  the  refraction  whose  logarithm  is  given  in  this 
table,  the  refraction  for  a  given  state  of  the  air  will  be  found  by 
the  formula 

r==  E^Y^ 

EXAMPLE  1. — Given  the  apparent  zenith  distance  z  =  78° 
30'  0",  Barom.  29.770  inches,  Attached  Therm.  —  0°.4  F.,  Ex- 
ternal Therm.  —  2°.0  F. 

We  find  from  Table  II.,  Col.  A,  for  78°  30', 

log  a  =  1.74981  A  =  1.0032  A  =  1.0328 

and  from  the  tables  for  barometer  and  thermometer, 


*  Tabulse  Regiomontanx,  p.  539. 


REFRACTION.  133 

log  B  =  4-  0.00253  log  r  =  4-  0.04545 

log  T=  4-  0.00127 
log  p  =  4-  0.00380 

Hence  the  refraction  is  computed  as  follows : 

log  a  =      1.74981 

Alogjl  =  log  ^  =  4-  0.00381 

A  log  r  =  log  ?-*  =  4-  0.04694 

log  tan  z  ==       0.69154 

r  =  310".53  =  5'  10".53    log  r  =       2.49210 

The  true  zenith  distance  is,  therefore,  78°  30'  0"  4-  5'  10".53  = 
78°  35'  10".53. 

EXAMPLE  2. — Given  the  true  zenith  distance  f  =  78°  35' 
10".53,  Barom.  29.770  inches,  Attached  Therm.  --  0°.4  F., 
External  Therm.  —  2°.0  F. 

We  find  from  Table  II.,  Col.  B,  for  78°  35'  10", 

log  a'  =  1.74680  A'  =  0.9967  A'  ==  1.0261 

and  from  the  tables  for  barometer  and  thermometer,  as  before, 

log  B  =  4-  0.00253  log  r  '=  4-  0.04545 

log  T=  4-  0.00127 

log  ft  =4-  0.00380  f 

The  refraction  is  then  computed  as  follows : 

log  a!  =       1.74680 

A'logfi  =  logpA'=:  +  0.00379 

X'  log  Y  =  log  Y*  =  +  0.04663 

log  tan  C  =       0.69489 

r  =  310".53"  =  5'  10".53  log  r  =       2.49211 

and  the  apparent  zenith  distance  is  therefore  78°  30'. 

EXAMPLE  3. — Given  z  =  87°  30',  barometer  and  thermometer 
as  in  the  preceding  examples. 

By  the  supplementary  table  above  given, 

log  £  =       2.98269 

A  =  1.0298      log  0  =  +  0.00380        log  p*  =  4-  0.00391 

A  =  1.2624      log  r  =  4-  0.04545        log  r*  =  +  0.05738 

r=      18'26".6       logr    =       3.04398 


134  REFRACTION. 

It  is  important  in  all  cases  where  great  precision  is  required 
that  the  barometer  and  thermometer  be  carefully  verified,  to  see 
that  they  give  true  indications.  The  zero  points  of  thermo- 
meters are  liable  to  change  after  a  certain  time,  and  inequalities 
in  the  bore  of  the  tube  are  not  uncommon.  A  special  investi- 
gation of  every  thermometer  is,  therefore,  necessary  before  it  is 
applied  in  any  delicate  research.  If  the  capillarity  of  the  baro- 
meter has  not  been  allowed  for  in  adjusting  the  scale,  it  must  be 
taken  into  account  by  the  observer  in  each  reading. 

We  may  obtain  the  true  refraction  for  any  state  of  the  air 
within  V  or  2",  very  expeditiously,  by  taking  the  mean  refrac- 
tion from  Table  I.  and  correcting  it  by  Table  XIV.  A,  and  Table 
XIV.  B.  The  mode  of  using  this  table  is  obvious  from  its 
arrangement.  Thus,  in  Example  1  we  find 

from  Table    I.,  Mean  refr.  =    4'  38".9 

"  XIV.  A,  for  Barom.  29.77,  Corr.  =  -      2  . 
"XIV.B,  "    Therm. —  2°.     «      =  -f  32  . 
True  refr.  =   5'     9". 

which  agrees  with  BESSEL'S  value  within  1".5.  For  greater 
accuracy,  the  height  of  the  barometer  should  be  reduced  to  the 
temperature  32°  F.,  which  is  the  standard  assumed  in  these 
tablep.  The  corrected  height  of  the  barometer  in  this  example 
is  29.85,  and  the  corresponding  correction  of  the  refraction 
would  then  be  —  1";  consequently  the  true  refraction  would  be 
5'  10",  which  is  only  0">5  in  error. 

These  tables  furnish  good  approximations  even  at  great 
zenith  distances.  Thus,  we  find  by  them,  in  Example  3,  r  = 

18'  24". 

» 

108.  INVESTIGATION  OF  THE  REFRACTION  FORMULA. — In  this 
investigation  we  may,  without  sensible  error,  consider  the  earth 
as  a  sphere,  and  the  atmosphere  as  composed  of  an  infinite 
number  of  concentric  spherical  strata,  whose  common  centre  is 
the  centre  of  the  earth,  each  of  which  is  of  uniform  density,  and 
within  which  the  path  of  a  ray  of  light  is  a  straight  line.  Let  C, 
Fig.  16,  be  the  centre  of  the  earth,  A  a  point  of  observation  on 
the  surface;  CAZ  the  vertical  line  ;  Aaf ,  a'b',  b'e',  &c.  the  vertical 
thicknesses  of  the  concentric  strata;  Se  a  ray  of  light  from  a  star 
S,  meeting  the  atmosphere  at  the  point  e,  and  successively  re- 


REFRACTION.  135 

fracted  in  the  directions  ed,  do,  &c.  to  the  point  A.  The  last 
direction  of  the  ray  is  aA,  which,  when  the  number  of  strata  is 
supposed  to  be  infinite,  becomes  a  tangent  to  the  curve  ecA  at  A, 
and  consequently  AaS'  is  the  apparent  direction  of  the  star.  Let 
the  normals  Ge,  Cd,  £c.  be  drawn  to  the  successive  strata.  The 
angle  Sef  is  the  first  angle  of  incidence,  the  angle  Ced  the  first 
angle  of  refraction.  At  any  intermediate  point  between  e  and  A, 
as  c,  we  have  Ccd,  the  supplement  of  the  angle  of  incidence,  and 
Ccb,  the  angle  of  refraction. 
If  now  for  any  point,  as  <?,  in  the  path  of  the  ray,  we  put 

i  =  the  angle  of  incidence, 
/  =  the  angle  of  refraction, 

fi.  =  the  index  of  refraction  for  the  stratum  above  c, 
I*'  =  "  "  "        below  c, 

then,  Art.  105, 

5E_*^  (148) 

sin/       n 
If  we  put 

q  =  the  normal  Cc  to  the  upper  of  the  two  strata, 
qf  =  "        Cb     "        lower         "  « 

i'  =  the  angle  of  incidence  in  the  lower  stratum, 
=  180°  —  Cbc, 

the  rectilinear  triangle  Cbc  gives 

sin  i' q 

sin  f~~<f 

which,  with  the  above  proportion,  gives 
q  fi  sin  i  =  q'f/  sin  i' 

an  equation  wrhich  shows  that  the  product  of  the  normal  to  any 
stratum  by  its  index  of  refraction  and  the  sine  of  the  angle  of 
incidence  is  the  same  for  any  two  consecutive  strata ;  that  is,  it 
is  a  constant  product  for  all  the  strata.  If  then  we  put 

z  =  the  apparent  zenith  distance, 

a  =  the  normal  at  the  observer,  or  radius  of  the  earth, 

At0  —  the  index  of  refraction  of  the  air  at  the  observer, 

we  have,  since  z  is  the  angle  of  incidence  at  the  observer, 

qn  sin  i  =  a/jt0  sin  z  (149) 


136  REFRACTION. 

in  which  the  second  member  is  constant  for  the  same  values  of 
z  and  /v 

.N"ow,  we  have  from  (148) 

tan  *  (i  -/)  =  ^=A  tan  J  (i  +  /) 

X  H~  P 

But  i  — /  is  the  refraction  of  the  ray  in  passing  from  one  stratum 
into  the  next ;  and  supposing,  as  we  do,  that  the  densities  of  the 
strata  vary  by  infinitesimal  increments,  i  —f  is  the  differential  of 
the  refraction  ;  and  we  may,  therefore,  write  J  dr  for  tan  J  (i  — /) 
and  dp.  for  p.'  —  p. ;  consequently,  also,  2p.  for  p.'  -j-  p,  and  tan  i  for 
tan  J  (i  +/) :  hence  we  have 

dr  =  ^  tan  i  (150) 

which  is  the  differential  equation  of  the  refraction. 

But,  as  both  p  and  i  are  variable,  we  cannot  integrate  this 
equation  unless  we  can  express  i  as  a  function  of  p.  This 
we  could  do  by  means  of  (149)  if  the  relation  between  q  and 
p  were  given,  that  is,  if  the  law  of  the  decrease  of  density  of  the 
air  for  increasing  heights  above  the  surface  of  the  earth  were 
known.  This,  however,  is  unknown,  and  we  are  obliged  to 
make  an  hypothesis  respecting  this  law,  and  ultimately  to  test 
the  validity  of  the  hypothesis  by  comparing  the  refractions  com- 
puted by  the  resulting  formula  with  those  obtained  by  direct 
observation.  I  shall  first  consider  the  hypothesis  of  BOUGUER, 
both  on  account  of  the  simplicity  of  the  resulting  formula  and 
of  its  historical  interest.* 

109.  First  hypothesis. — Let  it  be  assumed  that  the  law  of  de- 
crease of  density  is  such  that  some  constant  power  of  the  refrac- 
tion index  p  is  reciprocally  proportional  to  the  normal  q,  an 
hypothesis  expressed  by  the  equation 

*  I  shall  consider  but  two  hypotheses  :  the  first,  because  it  leads  to  the  simple 
formula  of  BRADLEY,  which,  though  imperfect,  is  often  useful  as  an  approximate 
expression  of  the  refraction;  the  second,  because  the  tables  formed  from  it  by 
BESSEL  have  thus  far  appeared  to  be  the  most  correct  and  in  greatest  accordance  with 
observation,  although  on  theoretical  grounds  even  the  hypothesis  of  BESSEL  is  open 
to  objection.  For  a  review  of  the  labors  of  astronomers  and  physicists  upon  this 
difficult  subject,  from  the  earliest  times  to  the  present,  see  Die  Astronomische  Strahlen- 
brechung  in  ihrer  historischen  Entwickelung  dargestellt,  von  DR.  C.  BRUHNS.  Leipzig, 
1861. 


FIRST   HYPOTHESIS.  137 

fcf-r         '        <"" 

which  with  (149)  gives 

(a  \n 
—  )   sin  z  (152) 

n     I 

or,  logarithmically, 

/sin  z\ 
log  sin  i  =  n  log  fji  -j-  log  I  — —  I 

\      •    0         / 

where  the  last  term  is  constant.     By  differentiation,  therefore, 

di  dfi 

tan  i  p 

which  with  (150)  gives 

di 

dr  =  — 

n  * 

and,  integrating, 


To  determine  the  constant  (7,  the  integral  is  to  be  taken  from 
the  upper  limit  of  the  atmosphere  to  the  surface  of  the  earth. 
At  the  upper  limit  r  —  0  ;  and  if  we  put  &  =  the  value  of  i  at  that 
limit,  we  have 


At  the   lower  limit  the  value  of  r  is  the  whole   atmospheric 
refraction,  and  i  =  z:  hence 

r  =  -  +  C 
n 

Eliminating  the  constant,  we  have 


To  find  $,  we  have,  by  putting  jj.  =  1  in  (152),  since  the  density 
of  the  air  at  the  upper  limit  is  to  be  taken  as  zero, 

sin  z 
sin  *  =  -  (154) 


Having  then  found  /*0  at  the  surface  of  the  earth  and  suitably 


138  'REFRACTION. 

determined  w,  we  find  &  by  (154),  and  then  r  by  (153).     The  two 
equations  may  be  expressed  in  a  single  formula  thus  : 


If  -i/sinsVl 

=  -\z  —  sin       -    - 
nl  \  /*0"  /J 


(155) 


which  is  known  as  SIMPSON'S  formula,  but  is  in  fact  equivalent 
to  the  formula  first  given  by  BOUGUER  in  1729  in  a  memoir  on 
refraction  which  gained  the  prize  of  the  French  Academy. 
From  (154)  we  find 

sin  z  —  sin  #  _  PL*  —  1 

sin  z  -f-  sin  #        /*0n  -f-  1 
whence 

tan  J(*  —  0)  =  //°n~1  tan  *  fz  +  *) 

AC  +  1 

and,  reducing  by  (153), 

tan  -  r  =  ^—  ^  tan  (  z  -  -  r  }  (156) 

2          M0n  +  1         \          27 

which  is  equivalent  to  BRADLEY'S  formula.  If  we  are  content  to 
represent  the  refraction  approximately  by  our  formula,  we  can 
write  this  in  the  form 

r  =  g  tan  (z  —  /r) 

and  we  shall  find,  with  BRADLEY,  that  for  a  mean  state  of  the  air 
corresponding  to  the  barometer  29.6  and  thermometer  50°  Fahr. 
we  can  express  the  observed  refractions,  very  nearly,  by  taking 

,  ^  =  57".036,        /=3. 

110.  But,  as  we  wish  our  formula  to  represent,  if  possible,  the 
actual  constitution  of  the  atmosphere,  let  us  endeavor  to  test  the 
hypothesis  upon  which  it  rests.  In  order  to  correspond  with  the 
real  state  of  nature,  it  is  necessary  that  the  constitution  of  the  atmo- 
sphere which  the  hypothesis  invokes  should  not  only  agree  with  the 
observed  refraction,  but  also  with  the  height  of  the  barometer,  and  with 
the  observed  diminution  of  heat  as  the  altitude  of  the  observer  above  the 
earth's  surface  increases. 

The  discussion  of  the  formula  will  be  more  simple  if  we  sub- 
stitute the  density  of  the  air  in  the  place  of  the  index  of  refrac- 
tion. Put 

8Q  =  the  density  of  the  air  at  the  surface  of  the  earth, 

S  =  the  density  of  the  air  at  any  point  above  the  surface. 


FIRST    HYPOTHESIS.  139 

The  relation  between  8  and  //,  according  to  Optics,  is  expressed  by 

t*—I  =  4kd  (157) 

in  which  4  k  is  a  'constant  determined  by  experiment.     Accord- 
ing to  the  experiments  of  BIOT, 

4  A  ==  0.000588768 

Since  k  is  so  small  that  its  square  will  be  inappreciable,  we  may 
take 


(158) 
and,  consequently, 


and  (156)  becomes,  still  neglecting  A-2, 

6vp(^^^y^  v 

tan-r==n/tf0tan(*-    ~r\  (159) 

If  we  denote  the  horizontal  refraction,  or  that  for  z  =  90°,  by  r0, 
this  formula  gives 


tan       r0  ==  nA^0  cot      r0 


or  tan  -  r  == 


and,  putting  the  small  arc  —  r0  for  its  tangent, 


We  can  find  <50  from  the  observed  state  of  the  barometer  and 
thermometer  at  the  surface  of  the  earth,  so  that  in  order  to  com- 
pute the  horizontal  refraction  by  this  formula,  for  the  purpose 
of  comparing  it  with  the  observed  horizontal  refraction,  we  have 
only  to  determine  the  value  of  n. 
Let 

x  —  the  height  of  any  assumed  point  in  the  atmosphere  above 

the  surface  of  the  earth, 

d,  p,  g  =  the  density  and  pressure  of  the  air,  and  the  force  of  grav- 
ity, respectively,  at  that  point, 
'V  Pv  ffo  —  the  same  quantities  at  the  earth's  surface. 


140  REFRACTION. 

At  an  elevation  greater  than  x  by  an  infinitesimal  distance  dx, 
the  pressure  p  is  diminished  by  dp.  The  weight  of  a  column  of 
air  whose  height  is  dx,  density  S,  and  gravity^,  is  expressed  by 
gddx,  and  this  is  equal  to  the  decrement  of  the  pressure:  hence 
the  equation 

dp  =  —  g3dx 

By  the  law  of  gravity,  we  have 

2 

V^.ifo*^ 

and  hence 

dx 

dp  =  —  g 


(a  +  X? 

(161) 


Now,  in  the  hypothesis  under  consideration,  we  have 
a  /Mn  +  1       /!  +  4A-3VL+2 


or,  neglecting  the  square  of  k, 

-^  =  1-5 

a  -}-  x 

which  gives 


Hence 

dp  =  2  g0a(n  -f- 
Integrating, 

(162) 


no  constant  being  necessary,  since  p  and  3  vanish  together. 
To  compare  this  with  the  observed  pressure^,,  let 

I  =  the  height  of  a  column  of  air  of  the  density  <\  which  acted 
upon  by  the  gravity  g0  will  be  in  equilibrium  with  the  pres- 
sure pQj 

in  other  words,  let  I  be  the  height  of  a  homogeneous  atmosphere 
of  the  density  £0  which  would  exert  the  pressure^.  Then,  by 
this  definition, 

A  =  ?.V  (163) 


FIRST   HYPOTHESIS.  141 

which  with  (162)  gives 


(.«> 


At  the  surface  of  the  earth,  w^here  p  becomes  pQ  and  d  becomes 
£0,  this  equation  gives 

1  =  (n  +  1)  ~  .  H  (165) 

whence  i 

— 

n==kd~'~l 
and  this  reduces  the  expression  of  the  horizontal  refraction  (160)  to 

9  kfi 

-•> 

<^tx 

Taking  as  the  unit  of  density  the  value  of  <50  which  corre- 
sponds to  the  barometer  0.76  metres  and  thermometer  0°  C., 
we  have,  according  to  BIOT, 

4A'<50  =  0.000588768 

The  constant  I  for  this  state  of  the  air  is  the  height  of  a  homo- 
geneous atmosphere  which  would  produce  the  pressure  Om.76  of 
the  barometer  when  the  temperature  is  0°  C. ;  and  this  height  is 
to  that  of  the  barometric  column  as  the  density  of  mercury  is  to 
that  of  the  air.  According  to  REGNAULT,  for  Barom.  0'n.76  and 
Therm.  0°  C.,  mercury  is  10517.3  times  as  heavy  as  air :  hence 
we  have 

I  =  0^.76  X  10517.3  =  7993-.15 

For  a  we  shall  here  use  the  mean  radius  of  the  earth,  since  we 
have  supposed  the  earth  to  be  spherical,  or 

a  =  6366738  metres 
which  gives 

l-  =  0.00125545  (167) 

Substituting  these  values  in  (166),  we  find,  after  dividing  by 
sin  1"  to  reduce  to  seconds, 

r0  =  1824"  =  30'  24" 
But,  according  to  ARGELANDER'S  observations,  we  should  have 


142  REFRACTION. 

for  Barom.  Om.76  and  Therm.  0°  C.,  rQ=  37'  31";  and  the  hypothesis 
therefore  gives  the  horizontal  refraction  too  small  by  more  than  V. 

111.  The  hypothesis  can  he  tested  further  by  examining 
whether  it  represents  the  law  of  decreasing  temperatures  for 
increasing  heights  in  the  atmosphere.  In  the  first  place,  we 
observe  that  in  this  hypothesis  the  densities  of  the  strata  of  the  atmo- 
sphere decrease  in  arithmetical  progression  when  the  altitudes  increase 
in  arithmetical  progression.  For,  since  x  is  very  small  in  compari- 
son with  #,  we  have  very  nearly 


a  -\-  x 
and  hence 

u*t«J  ^-  =  2  (w  +  : 

or,  by  (165), 

^-Ax/ly^v-vi)  x  =  21 

which  shows  that  equal  increments  of  x  correspond  to  equal 
decrements  of  S. 

This  last  equation  also  gives  for  the  upper  limit  of  the  atmo- 
sphere, where  d  —  0,  x  =  2  I ;  that  is,  in  this  hypothesis  the  height  of 
the  atmosphere  is  double  that  of  a  homogeneous  atmosphere  of  the  same 
pressure. 

Again,  we  have,  by  (164),  (165),  and  (168), 

tt^i'^1-*  (169) 

„      ^  *  O  7 

The  function  ^  expresses  the  law  of  heat  of  the  strata  of  the 

atmosphere.  For  let  r0  be  the  temperature  at  the  surface  of  the 
earth,  r  the  temperature  at  the  height  x.  If  the  temperature 
were  r0  in  both  cases,  we  should  have 

£-T  ("0) 


P 


but  when  the  temperature  is  changed  from  r0  to  T  the  density  is 
diminished  in  the  ratio  1  -f  e  (r  —  r0) :  1,  e  being  a  constant  which 


SECOND    HYPOTHESIS.  143 

is  known  from  experiment;  so  that  the  true  relation  between 
the  pressures  and  densities  at  different  temperatures  is  expressed 
by  the  known  formula 


whence 


which  combined  with  (169)  gives 


and  hence  equal  increments  of  x  correspond  to  equal  decrements 
of  r.  Hence,  in  this  hypothesis,  the  heat  of  the  strata  of  the  atmo- 
sphere decreases  as  their  density  in  arithmetical  progression.  The 
value  of  e,  according  to  RUDBERG  and  REGNAULT,  is  very  nearly 

1  2  1 

—  .     Hence  we  must  ascend  to  a  height  —  =  58.6  metres,  in 

order  to  experience  a  decrease  of  temperature  of  1°  C.  But, 
according  to  the  observations  of  GAY  LUSSAC  in  his  celebrated 
balloon  ascension  at  Paris  (in  the  year  1804),  the  decrease  of 
temperature  was  40°.  25  C.  for  a  height  of  6980  metres,  or  1°  C. 
for  173  metres,  so  that  in  the  hypothesis  under  consideration 
the  height  is  altogether  too  small,  or  the  decrease  of  temperature 
is  too  rapid.  This  hypothesis,  therefore,  is  not  sustained  either 
by  the  observed  refraction  or  by  the  observed  law  of  the  decrease 
of  temperature. 

112.  Second  hypothesis.  —  Before  proposing  a  new  hypothesis, 
let  us  determine  the  relation  between  the  height  and  the  density 
of  the  air  at  that  height,  when  the  atmosphere  is  assumed  to  be 
throughout  of  the  same  temperature,  in  which  case  we  should 
have  the  condition  (170).  Resuming  the  differential  equation 
(161), 


put 


144  REFRACTION. 

in  which  s  is  a  new  variable  very  nearly  proportional  to  x.     We 
then  have 

dp  =  —  gtiadds 

which  with  the  supposition  (170)  gives 


P  Po 

Integrating, 


in  which  the    logarithm  is    Napierian.     The    constant    being 
determined  so  that  p  becomes  p0  when  5  =  0,  we  have 


and  therefore 


P  < 

log  £  =  —  ^  as  =  --  = 

&#>  Po  I 


where  I  has  the  value  (163).     Hence,  e  being  the  Napierian  base, 

P  =  '  =  e-T  (172) 

Po        do 

which  is  the  expression  of  the  law  of  decreasing  densities  upon 
the  supposition  of  a  uniform  temperature.  In  our  first  hypo- 
thesis the  temperatures  decrease,  but  nevertheless  too  rapidly. 
We  must,  then,  frame  an  hypothesis  between  that  and  the  hypothesis  of 
a  uniform  temperature. 

Now,  in  our  first  hypothesis  we  have  by  (169),  within  terms 
involving  the  second  and  higher  powers  of  s, 

f^  _  l  _  — 
v  PoS  ~  21 

and  in  the  hypothesis  of  a  uniform  temperature, 


The  arithmetical  mean  between  these  would  be 

as 


SECOND    HYPOTHESIS.  145 

but,  as  we  have  no  reason  for  assuming  exactly  the  arithmetical 
mean,  BESSEL  proposes  to  take 


h  being  a  new  constant  to  be  determined  so  as  to  satisfy  the  observed 
refractions.  This  equation,  which  we  shall  adopt  as  our  second 
hypothesis,  expresses  the  assumed  law  of  decreasing  tempe- 
ratures, since,  by  (171),  it  amounts  to  assuming 

1  +  e  (r  -  TO)  =  <T?  (174) 

and  it  follows  that  in  this  hypothesis  the  temperatures  will  not 
decrease  in  arithmetical  progression  with  increasing  heights, 
though  they  will  do  so  very  nearly  for  the  smaller  values  of  s, 
that  is,  near  the  earth's  surface. 

Now,  combining  the  supposition  (173)  with  the  equation  /       *y 


l          =      -         y- 

we  have  //  ,   / 

dp  gj&  a±J  a   «. 

-*-  =  —  u-  °-ehds  =  --  eh  ds        -^  ^  *- 

PP.  It 

^      , 
Integrating  and  determining  the  constant  so  that  for  s  =  0,  p 

becomes  pOJ  we  have  '"»  &**•{*. 

2i  ;  =  -A     . 

-^iJ 


which  with  (173)  gives* 


Inasmuch  as  the  law  of  the  densities  thus  expressed  is  still 
hypothetical,  we  may  simplify  the  exponent  of  e.  For  if  h  is 
much  greater  than  I  (as  is  afterwards  shown),  we  may  in  this  ex- 


as 


ponent  put  e  h  —  1  =  —  and  we  shall  have  as  the  expression 
of  our  hypothesis 

_OS,OS  _*nl.«£  /"ITKN 

8  =  <50  e     i     h  ==  f50  e       h      i  (175) 

*  BESSEL.     Fundamenta  Astronomiae,  p.  28. 
VOL.  L— 10 


146  REFRACTION. 

By  comparing  this"  with  (172),  we  see  that  this  new  hypothesis 
differs  from  that  of  a  uniform  temperature  by  the  correction  - 

h 

applied  to  the  exponent  of  e. 
Putting,  for  brevity, 


we  have 

4  =  <50  e~*  (177) 


in  which  (3  is  constant.     This  expression  of  the  density  is  to  be 
introduced  into  the  differential  equation  of  the  refraction  (150). 
Now,  by  (149),  in  which  q  =  a  -{-  x,  we  have 

.     .  _     a/jLQ  sin  z     _  (1  —  s)  ^0  sin  z 

(a  -\-  X)[JL  fj. 

whence 

sin  i                          (1  —  s)  sin  z 
tan  i  = = '- 


(1  —  s)  sin 


^    cos2  z  —  (  1  — ~  j+  (25  —  s2)  sin2 


From  the  equation  jtt2  =  1  +  4  kd  we  deduce 

2kdd 


and  if  we  introduce  as  a  constant  the  quantity 

(178) 


1  -f  4#<50 
(which  for  Barom.  Om.76  and  Therm.  0°  C.  is  a  =  0.000294211) 


1  — 2a(l  — - 
"We  might  neglect  the  square  of  &,  and  consequently,  also,  that  of 


SECOND    HYPOTHESIS.  147 

a,  with  hardly  appreciable  error,  and  then  this  expression  would 
become  simply  a  —  -  ,  but  for  greater  accuracy  we  can  retain  the 

denominator,  employing  its  mean  value,  as  it  varies  within  very 
narrow  limits.  For  its  greatest  value,  when  d  —  £0,  is  =  1, 
and  its  least  value,  when  d  —  0,  is  =  1  -  -  2oc,  and  the  mean 
between  these  values  is  1  —  a.  Hence  we  shall  take 

dp.  a         dd 


In  the  denominator  of  the  value  of  tan  i  we  have  also  to  sub- 
stitute 


fj.\  1  -f  4A'(50 

Therefore,  substituting  in  (150),  we  have 

a  sin  z  (1  —  s)  - 


(1  —  a)  [cos2  g  —  2a  (  1  —  —  \  +  (2s  —  s')  sin2  z]  * 


or,  by  (177), 

—  a/9  sin  2  (1  —  s) 
dr  —  -  - 


(1  —  a)  [cos2  2  —  2a(l  —c-^)  +(25—  s2)  sin2z]* 

In  the  integration  of  this  equation  we  may  change  the  sign  of 
the  second  member,  since  our  object  is  only  to  obtain  the 
numerical  value  of  r.  It  is  apparent  that  if  we  put  1  for  1  —  s 
in  the  numerator  of  this  expression,  and  also  neglect  the  term 
s2sin2£  in  the  denominator,  the  error  will  be  almost  or  quite 
insensible;  but,  not  to  reject  terms  without  examination,  let  us 
develop  the  expression  into  series.  For  this  purpose,  put  the 
radical  in  the  denominator  under  the  form  i/  \f  —  s2  sin2  z9  in 
which 

y  =  [cos2  z  —  2a  (1  —  e-Ps)  +  2s  sin2  zf 
Then 


l  —  s 


148  REFRACTION. 

1        2s?/2  —  s2sin22 

Hen  ce,  restoring  the  value  of  ?/,  we  have 

a/3  sin  z  e 


(1  —  a)  [cos2  z  —  2a  (1  —  e~^  -f  2s  sin2  /J2 

a/9  sin  z  e~^ssds  [cos2  z  —  2a  (1  —  e~ps)  -f-  -f  s  sin2  z] 
(1  _  a)  [Cos2  2  —  2a  (1  —  e-**)  +  2s  sin2/]! 

-&c (179) 

We  shall  hereafter  show  that  the  second  term  of  this  develop- 
ment is  insensible.  Confining  ourselves  for  the  present  to  the 
first  term,  let  us,  after  the  method  of  LAPLACE,  introduce  the  new 
variable  s'  such  that 

s  =  s'  +  ^-^>  (180) 

4.V<H  sin'2 

then  this  term  takes  the  form 

dr  =  a/98in^      "* -,  (181) 

(1  —  a)  [cos2  z  -f  2s'  sin2  z}% 

in  which  we  have  yet  to  reduce  the  numerator  to  a  function  of 
the  new  variable  s'.  Now,  b}7  Lagrange's  Theorem,*  when 

*  See  PEIRCE'S  Curves  and  Functions,  Vol.  I.  Art.  181.  For  the  convenience  of 
the  reader,  however,  I^add  the  following  demonstration  of  this  theorem.  It  is  pro- 
posed to  develop  the  function  u  =fy  in  a  series  of  ascending  powers  of  x,  x  and  y 
being  connected  by  the  equation 

y  =  t  4-  z<t>y 

and  the  functions/and  0  being  given.  If  from  this  equation  y  could  be  found  as  an 
explicit  function  of  x  and  substituted  in  the  equation  u  =fy,  the  development  could 
be  effected  at  once  by  Maclaurin's  Theorem,  according  to  which  we  should  have 

u  =  uQ  -f-  D  UQX  -|-D2M0 -f-....-f  D  n  UQ  —  —  -f-  &c. 

1.2  1  .'  2  .  .  .  .  n 

where  u0,  DxuQ,  &c.  denote  the  values  of  u  and  its  successive  derivatives  when  x  =  0. 
It  is  proposed  to  find  the  values  of  the  derivatives  without  recourse  to  the  elimination 
of  ?/,  as  this  elimination  is  often  impracticable.  For  brevity,  put  Y  =  qy ;  then  the 
derivatives  of 

relatively  to  x  and  t  are 


SECOND    HYPOTHESIS.  149 

S  =  Sf  -\-  a,<f>S 


we  have 

ft  =  #+•$-  |>'.  Dfs'  ]  +  ±  D 


in  which  /  and  ^  denote  any  functions  whatever,  and  D,  D2,  £c. 
the  successive  derivatives  of  the  functions  to  which  they  are 
prefixed.  Hence,  by  putting 

1  —  P-** 
- 


sin2  z 
this  theorem  gives 


— - 


Dxy  =  Y  +  xDyYDxy  Dty  =  1  +  x  DyYDt  y 

whence,  eliminating  x, 

Dxy=YDty 

Multiplying  this  by  Dyu,  it  gives 

Dxu=YDtu  (a) 

The  derivative  of  this  equation  relatively  to  t  is 


This  is  a  general  theorem,  whatever  function  u  is  of  y,  and  consequently,  also,  what- 
ever function  Dtu  is  of  y.  We  may  then  substitute  in  it  the  function  YnDtu  for  Dtu, 
and  we  shall  have 

(6} 


Now,  the  successive  derivatives  of  (a)  relatively  to  x  are,  by  the  successive  appli- 
cation of  (6),  making  n  =  1,  2,  3,  &c., 

D*  u  =  Dx[  YDtu-\     =  Dt 
D*  u  =  D 


But  when  x  =  0,  we  have  y  =  t,  Y=  <j>t,  and  hence 


where  the  subscript  letter  of  the  D  is  omitted  in  the  second  members  as  unnecessary, 
since  t  is  now  the  only  variable.  These  values  substituted  in  Maclaurin's  Theorem 
give  Lagrange's  Theorem  :  , 


150  REFRACTION. 


1.28in42 

a3  ft 

1.2.3  sin62 


1.  2.  8...  n  sin**  2| 

-  &c.  (182) 

But  we  have  in  the  numerator  of  (181) 


and  hence,  differentiating  (182)  and  substituting  the  result  in 
(181),  we  find 

dr= 


—  a)  [cos2  2  +  2s'  sin2  z 

D2  [(1  -  e-*s'y  e-fi' 
LV 


+  &c.  1  (183) 

To  effect  the  differentiations  expressed  in  the  several  terms  of 
this  series,  we  take  the  general  expression 


where  the  upper  sign  is  to  he  used  when  n  is  even,  and  the  lower 
sign  when  n  is  odd;  Differentiating  this  n  •.  times  successively, 
we  have 


SECOND   HYPOTHESIS.  151 

by  means  of  which,   making  n  =  1  .  2  .  3  .  .  .  successively,   we 
reduce  (183)  to  the  following  form  : 

•    Bin  ads' 


(1  —  a)  [cos2  z  +  2  s'  sin2  2]*  I  sin2  z 


(184) 

We  have  now  to  integrate  the  terms  of  this  series,  after  having 
multiplied  each  by  the  factor  without  the  brackets.  The  inte- 
grals are  to  be  taken  from  the  surface  of  the  earth,  where  s  =  0, 
to  the  upper  limit  of  the  atmosphere ;  that  is,  q  being  the  nor- 
mal to  any  stratum  (Art.  108),  they  are  to  be  taken  between  the 
limits  q  =  a  and  q  =  a  -f  H9  H  being  the  height  of  the  atmo- 
sphere. Now,  this  height  is  not  known ;  but  since  at  the  upper 
limit  the  density  is  zero  and  beyond  this  limit  the  ray  suffers 
no  refraction  to  infinity,  we  can  without  error  take  the  integrals 
between  the  limits  q  =  a  and  q  =  oo  ,  i.e.  between  5  =  0  and 
5  =  1.  But  we  may  make  the  upper  limit  of  5  also  equal  to  in- 
finity. For,  by  (176),  /9  will  not  differ  greatly  from  -,  and  conse- 
quently will  be  a  very  large  number,  nearly  equal  to  800,  as  we 
find  from  (167) ;  hence  for  s  =  1  we  have  in  (172)  3  =  ° — - 

which  will  be  sensibly  equal  to  zero,  and  consequently  the  same 
as  we  should  find  by  taking  5  =  oo  .     Hence  the  integrals  may 
be  taken  between  the  limits  5  =  0  and  s  =  oo  ;  consequently, 
also,  according  to  (180),  between  the  limits  5'  =  0  and  5'  =  oo  . 
Now,  as  every  term  of  the  series  will  be  of  the  form 

/?  sin  2  ds'e~nf*sf  0ds'e-nf*s' 

(loo) 


[cos2  2  -f  2  s'  sin2  zfi       [cot2  z  -f  2  s'lfi 

multiplied  by  constants,  we  have  only  to  integrate  this  general 
form.     Let  t  be  a  new  variable,  such  that 

9/a 

cot2 2 -f  2s' =      -  (186) 


152  REFRACTION. 

then  (185)  becomes 


the  integral  of  which  is  to  be  taken  from  t  = 

T  (187) 


to  t  —  <x>,  which  are  the  limits  given  by  (186)  for  sf  =  0  and 
6'  =  oo  .     If,  therefore,  we  denote  by  ^  (n)  a  function  such  that 


or 

4f(ri)  =  eTTf'"dte-tt  (188) 

*/    JT 

the  integral  of  (185)  will  become 

f     '*'"""""'".  -y^.iff  (189) 

^  2  '2 


Substituting  this  value  in  (184),  making  successively  ft  =  1,  2,  3, 
&c.,  we  find  the  following  expression  of  the  refraction: 


r  = 


~  2* 


-8     3 


+  &c.  (190) 

which,  since  we  have  in  general 


1        1.2       1  .  2.3 


SECOND   HYPOTHESIS. 

can  also  be  written  as  follows  :* 


153 


•1/20 

^             a  ft 

•40 

e    «taa«  4(2) 

""                 sin2  2 

3 

32?                  .,2  /O2 
o,  p 

I                           ,              ' 

e     •«»»«  4(8) 

4a/J        - 

e     »^«  4(4) 

1     1.2       sin** 

41             a3/?3 

'   1.2.3     sin6* 
+  &c. 

(191) 


113.  The  only  remaining  difficulty  is  to  determine  the  func- 
tion $(n),  (188).  In  the  case  of  the  horizontal  refraction,  where 
cot  z  =  0  and  therefore  also  T  =  0,  this  function  becomes 
independent  of  (w),  and  reduces  to  the  well-known  integral! 


dte-tl  = 


(192) 


*  LAPLACE,  Mecanique  Celeste,  Vol.  IV.  p.  186  (BOWDITCH'S  Translation) ;  where, 
however,  -  stands  in  the  place  of  the  more  general  symbol  (3  here  employed.  This 
form  of  the  refraction  is  due  to  KRAMP,  Analyse  des  refractions  astronomiques  et  ler- 
restres,  Strasbourg,  1799. 

f  This   useful  definite   integral   may  be  readily  obtained   as  follows.     Put  k  = 

\      dt  e  —  tt.     Then,  since  the  definite  integral  is  independent  of  the  variable,  we 
•Jo 

/•oo 

also  have  k  =(     do  e  ~~  DV  ,  and,  multiplying  these  expressions  together, 


the  order  of  integration  being  arbitrary.     Let 

v  =  tu ;  whence  dv  =  t  du 
(for  in  integrating,  regarding  v  as  variable,  £  is  regarded  as  constant) :  then  we  have 

k*=  f*  f^rfu  .  dt.  te~ tt(l  +  uu)—  C     du]      dt.te—u(l+™) 
J o  ^o  ^o       ^o 


1  i          _x  __  x       TT 

— — —  —  y  (tan        oo  -  J  —  - 


whence 


154 


REFRACTION. 


where  n  =  3.1415926  ....  The  expression  for  the  horizontal 
refraction  is  therefore  found  at  once  by  putting  JI/TT  for  $  (n) 
in  every  term  of  (191),  and  sin  2  =  1,  namely: 


1  — 


(193) 


&C. 


For  small  values  of  T,  that  is,  for  great  zenith  distances,  we 
may  obtain  the  value  of  the  integral  in  (188)  by  a  series  of 
ascending  powers  of  T.  We  have 


/GO  X»QO  /V 

#«-"*=*  I     #«-"— I 
r  Vo  */  n 


die" 


(194) 


The  first  integral  of  the  second  member  is  given  by  (192).     The 
second  is 


7"*  3 

=  T—  — 
3 


1  775 

-1-  .  - 
1.2      5 


1.2.3     7 


.  —  +  &c. 


(195) 


Another  development  for  the  same  case  is  obtained  by  the  suc- 
cessive application  of  the  method  of  integration  by  parts,  aa 
follows:* 


-"  =  t  e-il 


die 


—  it 


*  By  the  formula  fxdy  =  xy  — fydx,  making  always  x  =  e     ",  and  cfy  succes- 
sively =  dt,  t*dt,  t\dt,  &c. 


SECOND    HYPOTHESIS.  155 


3         3.5        3.5.7 
whence,  by  introducing  the  limits, 

2  T2      (2  T2N*2       f2  T 


As  the  denominators  increase,  these  series  finally  become  con- 
vergent for  all  values  of  T;  but  they  are  convenient  only  for 
small  values. 

For  the  greater  values  of  T,  a  development  according  to  the 
descending  powers  may  be  obtained,  also  by  the  method  of 
integration  by  parts,  as  follows:*  We  have 


-"  =  -±-  e-tf  -  J  f— 
^t  ^J    p 


1 

=  --  e-ft-4 
2t 

Hence 


2T  2T2(2  T2)2       (2 

-jL       „ 


(2  T2)» 

The  sum  of  a  number  of  consecutive  terms  of  this  series  is 
alternately  greater  and  less  than  the  value  of  the  integral.  But 
since  the  factors  of  the  numerators  increase,  the  series  will  at 
last  become  divergent  for  any  value  of  T.  Nevertheless,  if  we 
stop  at  any  term,  the  sum  of  all  the  remaining  terms  will  be  less  than 
this  term;  for  if  we  take  the  sum  of  all  the  terms  in  the  brackets, 
the  sum  of  the  remaining  terms  is 


By  the  formulay"z  dy  =  xy  —  fy  dx,  making  always  dy  =  t  dt  e~ft  ,  and  z 


successively  =-,  —  »  -»  &c. 
J        t    t*  t* 


156  REFRACTION. 

The  integral  in  this  expression  is  evidently  less  than  the  product 
of  the  integral 


multiplied  by  the  greatest  value  of  e~lt  between  the  limits 

oo  ,  and  this  greatest  value  is  e~TT.     Hence  the  above  remainder 

is  always  numerically  less  than 

_1  .3.5  ....(2ft  —  1)    _rr 


which  expression  is  nothing  more  than  the  last  term  of  the  series 
(when  multiplied  by  the  factor  without.  the  brackets),  taken  with 
a  contrary,  sign.  Hence,  if  we  do  not  continue  the  summation 
until  the  terms  begin  to  increase,  but  stop  at  some  sufficiently 
small  term,  the  error  of  the  result  will  always  be  less  than  this 
term. 

Finally,  the  integral  may  be  developed  in  the  form  of  a  con- 
tinued fraction,  as  was  shown  by  LAPLACE.     Putting  for  brevity 

4<n)  =  ti0  =  —  (  1  --  -  +  -111  -  ^^  -f  &c.  \         (198) 
2T\         2T*      (2T2)2      (2T2/  / 

and  denoting  the  successive  derivatives  of  i/0  relatively  to  T  by 
ww  uv  &c.,  we  have  first 

c.  (199) 


l 

2T2       (2T2)2       (2T2)3 

or 

M1  =  2Z\i0  —  l  (200) 

Differentiating  this  equation,  successively,  we  have 


&c. 
or,  in  general, 


n  having  any  value  in  the  series  1.2.3.4...  &c. 


.    • 

&         of  ~f'^' 

fiiiiws 


SECOND   HYPOTHESIS.  157 

Hence  we  derive 


u 


2nl-:t 

(202) 


or,  putting 

*  =  (201) 


w»-i 
By  (200)  we  have 


or 

(203) 


But  from  (202),  by  making  n  successively  1,  2,  3,  &c.,  we  have 

2  (-^ 

&c., 


!  _  (D*  J  "*  i  _  (S)'  2 

which  successively  substituted  in  (203)  give 
1 


1  +  &c.  (204) 

This  can  be  employed  for  all  values  of  T,  but  when  k  exceeds  J 
it  will  be  more  convenient  to  employ  (195)  or  (196). 
The  successive  approximating  fractions  of  (204)  are 

1  1  1   _L  9b  1    J_  W  1    _J_     Q/'  J_     8£* 

1  -+-  LK  1    -T-   "/v  A    -j-      ilA  -f"      oA. 


l         1  +k         1-I-3A          l_f-6A'  +  3-       1  -f  10A;  +  15A:8 
and,  in  general,  denoting  the  ?i'/l  approximating  fraction  by  7^, 


158  REFRACTION. 

an       an-\  +  (n  —  1 


bn       bn-i  -f-  (7i  —  1)  A-6n_2 

• 

By  the  preceding  methods,  then,  the  function  $(ri)  can  be 
computed  for  any  value  of  T.  A  table  containing  the  logarithm 
of  this  function  for  all  values  of  T  from  0  to  10,  is  given  by 
BESSEL  (Fundamenta  Astronomies,  pp.  36,  37),  being  an  extension 
of  that  first  constructed  by  KRAMP.  By  the  aid  of  this  table  the 
computation  of  the  refraction  is  greatly  facilitated. 

>p'^ 

114.  Let  us  now  examine  the  second  term  of  (179.)  This  term 
will  have  its  greatest  value  in  the  horizontal  refraction,  when 
z  —  90°,  in  which  case  it  reduces  to 

a/3  e-*s  sds       s  —  2a  (1  —  «-*) 


(1  —  a)  [2<j  _  2a(l  —  C-*)] 

Moreover,  the  most  sensible  part  of  the  integral  corresponds  to 
small  values  of  s,  and  therefore,  since  a  is  also  very  small,  we 
may  put  2a  (1  —  e~fts)  =  2afts.  The  integral  thus  becomes 


2*(1  —  a)(l  —  o/9) 

Now  we  have,  by  integrating  by  parts, 

/ 
, 

and  hence 


-I  /»  00 

^==—  «- 

J 


Putting  /3s  —  x2,  this  becomes,  by  (192), 
I00 


2/5 


Hence  the  term  becomes 

q(3—  4  o/9)  /V 

8  (1—  a)(l  —  a/5)l   \2/9 


SECOND   HYPOTHESIS.  159 

Taking  BESSEL'S  value  of  h  =  =  116865.8  toises*  =  227775.7 
metres,  and  the  value  of  1  =  7993.15  metres  (p.  141),  we  find  by 
(176)  p  =  768.57.  Substituting  this  and  a  =  0.000294211  (p.  146), 
the  value  of  the  above  expression,  reduced  to  seconds  of  arc  by 
dividing  by  sin  1",  is  found  to  be  only  0".72,  which  in  the  hori- 
zontal refraction  is  insignificant.  This  term,  therefore,  can  be 
neglected  (and  consequently  also  all  the  subsequent  terms),  and 
the  formula  (191)  may  be  regarded  as  the  rigorous  expression  of  ' 
the  refraction. 

115.  In  order  to  compute  the  refraction  by  (191),  it  only  re- 
mains to  determine  the  constants  a  and  p.  The  constant  a 
might  be  found  from  (178)  by  employing  the  value  of  k  deter- 
mined by  BIOT  by  direct  experiment  upon  the  refractive  power 
of  atmospheric  air,  but  in  order  that  the  formula  may  represent 
as  nearly  as  possible  the  observed  refractions,  BESSEL  preferred 
to  determine  both  a  and  p  from  observations.f 

Now,  a  depends  upon  the  density  of  the  air  at  the  place  of 
observation,  and  is,  therefore,  a  function  of  the  pressure  and 
temperature;  and /?,  which  involves  /,  also  depends  upon  the  ther- 
mometer, since  by  the  definition  of  I  it  must  vary  with  the  tem- 
perature. The  constants  must,  then,  be  determined  for  some 
assumed  normal  state  of  the  air,  and  we  must  have  the  means 
of  deducing  their  values  for  any  other  given  state.  Let 

pQ  =  the  assumed  normal  pressure, 

r0  =        "  "  temperature, 

p  —  the  observed  pressure, 

r  —    "         "         temperature, 

30  =  the  normal  density  corresponding  to  p0  and  r0, 

3  =  the  density  corresponding  to^?  and  T; 

*  Fundamenta  Astronomise,  p.  40. 

f  It  should  be  observed  that  the  assumed  expression  of  the  density  (177)  may 
represent  various  hypotheses,  according  to  the  form  given  to  /?.  Thus,  if  we  put 

/3  =  -,  we  have  the  form  (172)  which  expresses  the  hypothesis  of  a  uniform  tem- 
perature. We  may  therefore  readily  examine  how  far  that  hypothesis  is  in  error  in 
the  horizontal  refraction;  for  by  taking  the  reciprocal  of  (167)  we  have  in  this  case 
3  =  796.53,  and  hence  with  a  =  0.00029411  we  find,  by  taking  fifteen  terms  of  the 
series  (193),  r0  =  39'  54".5,  which  corresponds  to  Barom.  Om.  76,  and  Therm.  0°  C. 
This  is  2'  23".5  greater  than  the  value  given  by  ARGELAXDER'S  Observations  (p.  141). 
Our  first  hypothesis  gave  a  result  too  small  by  more  than  7',  and  hence  a  true  hypo- 
thesis must  be  intermediate  between  these,  as  we  have  already  shown  from  a  con 


160  REFRACTION. 

then  we  have  by  (171) 


in  which  £  is  the  coefficient  of  expansion  of  atmospheric  air,  or 
the  expansion  for  1°  of  the  thermometer.  If  the  thermometer  is 
Centigrade,  we  have,  according  to  BESSEL,* 

e  ==  0.0036438 

From  (178)  it  follows  that  a  is  sensibly  proportional  to  the 
density,  and  hence  if  we  put 


a0  =  the  value  of  a  for  the  normal  density  <50, 
we  have,  for  any  given  state  of  the  air, 


(205) 


in  which  for  p  and  pQ  we  may  use  the  heights  of  the  barometric 
column,  provided  these  heights  are  reduced  to  the  same  tem- 
perature of  the  mercury  and  of  the  scales. 
Again,  if 

1Q  =  the  height  of  a  homogeneous  atmosphere  of  the  temperature 
TO  arid  any  given  pressure, 

then  the  height  I  for  the  same  pressure,  when  the  temperature 

is  r,  is 

The  normal  state  of  the  air  adopted  by  BESSEL  in  the  determi- 
nation of  the  constants,  so  as  to  represent  BRADLEY'S  observa- 
tions, made  at  the  Greenwich  Observatory  in  the  years  1750- 
1762,  was  a  mean  state  corresponding  to  the  barometer  29.6 
inches,  and  thermometer  50°  Fahrenheit  =  10°  Centigrade;  and 
for  this  state  he  found 

«„  =  0.000278953 


sideration  of  the  law  of  temperatures.  At  the  same  time,  we  see  that  the  hypothesis 
of  a  uniform  temperature  is  nearer  to  the  truth  than  the  first  hypothesis,  and  we  are 
so  far  justified  in  adhering  to  the  form  6  =  cJ0«-^  with  the  modification  of  substi- 
tuting a  corrected  value  of  3. 

*This  value,  determined  by  BESSEL,  from  the  observations  of  stars,  differs  slightly 
from  the  value  -%fa  more  recently  determined  by  RUDBERQ  and  REGNAULT  by  direct 
experiments  upon  the  refractive  power  of  the  air. 


SECOND    HYPOTHESIS.  161 

or,  dividing  by  sin  1", 


and 

h  =  116865.8  toises  —  227775.7  metres. 

For  the  constant  /0  at  the  normal  temperature  50°  F.,  BESSEL 

employed 

10  =  4226.05  toises  =  8236.73  metres.* 

Since  the  strata  of  the  atmosphere  are  supposed  to  be  parallel  to 
the  earth's  surface,  BESSEL  employed  for  a  the  radius  of  curva- 
ture of  the  meridian  for  the  latitude  of  Greenwich  (the  observa- 
tions of  Bradley  being  taken  in  the  meridian),  and,  in  accordance 
with  the  compression  of  the  earth  assumed  at  the  time  when 
this  investigation  was  made,  he  took 

a  =  6372970  metres. 
Hence  we  have 

ft  =  h—^  •  -  =  745.747 

These  values  of  a0  and  /90  being  substituted  for  a  and  /?  in 
(193),  the  horizontal  refraction  is  found  to  be  only  about  V  too 
great,  which  is  hardly  greater  than  the  probable  error  of  the 
observed  horizontal  refraction.  At  zenith  distances  less  than 
85°,  however,  BESSEL  afterwards  found  that  the  refraction  com- 
puted with  these  values^of  the  constants  required  to  be  multi- 
plied by  the  factor  1.003282  in  order  to  represent  the  Konigsberg 
observations. 

116.  By  the  preceding  formulae,  then,  the  values  of  the  con- 
stants a  and  /9  can  be  found  for  any  state  of  the  air,  as  given  by 
the' barometer  and  thermometer  at  the  place  of  observation,  and 
then  the  true  refraction  might  be  directly  computed  by  (191). 
But,  as  this  computation  would  be  too  troublesome  in  practice, 
the  mean  refraction  is  computed  for  the  assumed  normal  values 
of  a  and  /9,  and  given  in  the  refraction  tables.  From  this  mean 

*  According  to  the  later  determination  of  REONATLT  which  we  have  used  on  p.  143, 
we  should  have  l^=  8286.1  metres.     The  difference  does  not   affect   the  value  of 
BESSKJ/S  tables,  which  are  constructed  to  represent  actual  observations. 
Vor..  I.— 11 


162  REFRACTION. 

refraction  we  must  deduce  the  true  refraction  in  any  case  by 
applying  proper  corrections  depending  upon  the  observed  state 
of  the  barometer  and  thermometer.  For  facility  of  logarithmic 
computation,  BESSEL  adopted  the  form 


in  which  r0  is  the  tabular  refraction  corresponding  to  p0  and  TO, 
and  r  is  the  refraction  corresponding  to  the  observed  p  and  T. 
Let  us  see  what  interpretation  must  be  given  to  the  exponents 
A  and  L  If  the  pressure  remained  pw  the  refraction  correspond- 
ing to  the  temperature  r  would  be 


dr  dr*      1. 

or,  with  sufficient  precision, 

1     dr  , 


In  like  manner,  if  the  temperature  were  constant,  and  the  pres- 
sure is  increased  by  the  quantity  p  —  p0,  the  refraction  would 
become  nearly 


dp^ 

Hence,  when  both  pressure  and  temperature  vary,  we  shall  have, 
very  nearly, 

-f  i  •  ^  (r  —  TO)  |    (208) 

Now,  putting—  in  (207)  under  the  form  1  -f  -— — 5,  and  develop- 
ing by  the  binomial  theorem,  we  have 


Therefore,  neglecting  the  smaller  terms,  we  must  have 


.  (209) 

0     dr 


SECOND    HYPOTHESIS.  163 

to  determine  which  we  are  now  to  find  the  derivatives  of  (191) 
relatively  to  p  and  r.     Put 

(210) 


sin2  z 


and  ql  =  $  (1),  ft  =  2^(2),  ft  =  3*^(3),  &c.,  or  in  general 

2n  — 1 

(211) 


then,  if  we  also  put 

Q ~,  o  — x  Q    [      *      Q  —  2z  Q  i \ ^ — nx  ~    I    ^Q      f  212^ 

the  formula  (191)  becomes 

1-*  5 

(1  —  a)  r  =  sin2*  «|±.  §  (213) 

'  %/      -w 

/^ 

in  which,  since  the  variations  of  ~    -  in  (191)  are  sensibly  the 

same  as  those  of  a,  we  may  regard  1  —  a  as  constant.  Differen- 
tiating this,  observing  that  Q  varies  with  both  j?  and  r,  while  ft 
varies  only  with  r,  we  have 

dr  -     '  2       /2    dQ  ^ 

P    -  (214) 

In  differentiating  §,  it  will  be  convenient  to  regard  it  as  a  func- 
tion of  the  two  variables  x  and  /?,  the  quantities  ft,  ft,  &c.  vary- 
ing only  with  ft.  We  have,  since  ft  does  not  vary  with.^>, 

dO      dO    dx  /oi  K\ 

.  {£lOj 

dp       dx      dp 
and  since  both  x  and  8  vary  with  r, 

dO      dO     dx   ,   dO     dft  ,<\t-/*\ 

— -  ==  — —  • (-  - —  •  —  (*!") 

From  (212)  we  find 


164  REFRACTION. 


in  which 


.  ~  dp  dT      dp 

1^,17  «s>-/ 

But  by  (200),  in  which  w0  =  ^(n),  we  have 


and  by  (1ST) 


whence 


Also, 

^  =  xe-**  +  -*.  e  -  -  ^  +  &c.  (219) 

<*/?  d/3     r1.2  dS 

in  which  we  have  generally,  by  (211), 


^         /9     n       2/5 

cot2  ^  cot  z 

.==.  -  n  q  --    —  nn 

2  i/2j 

Substituting  the  values  of  this  expression  for  n  =  1,  2,  3,  &c.  in 
(219),  we  have 

3  to. 


1.2  1.2.3          ,. 


The  first  series  in  this  expression  —  §'.     The  second,  when 
e  ~  x,  e~2x,  &c.  are  developed  in  series,  becomes 


x  -f  x*  -f  a^  +  &c.  = 


1—  a? 


SECOND    HYPOTHESIS.  165 

and  hence 

^  =  cot2  z  Q*  __  tcoi^_  .     x  (220) 

We  have,  further,  from  (210)  and  the  values  of  a,  I,  and  ft  in  the 
preceding  article, 

dx  X      da         X      a        X 

dp         a,      dp         a-     P        P 

dfi        d,3      dl  a  h 

dr          dl       dr  I1  h  —  I 

,  1 , 4  , 
daL_  .v,^    oC>^V 

^7" 

<^.r  __  x    da      x    dj$  _               2  h  —  l 
dr       a,     dr       ft     dr  h / 

Substituting  these  values  in  (215)  and  (216),  and  then  substituting 
in  (214),  we  find* 


dp  /  p 


These  formulae  are  to  be  computed  with  the  normal  values  of  a, 
ft,  r,  I,  and  p,  and  for  the  different  zenith  distances,  after  which 
A  and  ^  are  computed  by  (209).  The  values  of  A  and  ^  thus 
found  are  given  in  Table  II. 

117.  Finally,  in  tabulating  the  formula  (207),  BESSEL  puts  . 

r0  =  a  tan  z  (222) 


Po  l  +  *  (T  —  To) 

(where  a  and  /9  no  longer  have  the  same  signification  as  in  the 
preceding  articles). 

*  BESSEL,  Fundamenta  Astronomix,  p.  34. 


16(3  REFRACTION. 

The  true  refraction  then  takes  the  form 

r  =  apAr*  tan  z  (223) 

The  quantity  here  denoted  by  /9  is  the  ratio  of  the  observed  and 
normal  heights  of  the  barometer,  both  being  reduced  to  the  same 
temperature  of  the  mercury  and  of  their  scales.  First,  to  correct 
for  the  temperature  of  the  scale,  let  b(l\  b(°\  or  b(m)  denote  the  ob- 
served reading  of  the  barometer  scale  according  as  it  is  graduated 
in  Paris  lines,  English  inches,  or  French  metres.  The  standard 
temperatures  of  the  Paris  line  is  13°  Reaumur,  of  the  English  inch 
62°  Fahrenheit,  and  of  the  French  metre  0°  Centigrade  ;  that  is, 
the  graduations  of  the  several  scales  indicate  true  heights  only 
when  the  attached  thermometers  indicate  these  temperatures 
respectively.  The  expansion  of  brass  from  the  freezing  point  to 
the  boiling  point  is  .0018782  of  its  length  at  the  freezing  point. 
If  then  the  reading  of  the  attached  thermometer  is  denoted 
either  by  r',  /',  or  c',  according  as  it  is  Reaumur's,  Fahrenheit's, 
or  the  Centigrade,  the  true  height  observed  will  be  (putting  s  = 
0.0018782) 

l+-r'  !+—(/'  —  32)  1+—  -c' 

80  *° 


-  .  13  1  +  —  •  30 

80  r  180 


or 


SOj-r^  (e)   18Q  +  (/'-32)s  IQQ+c'j 

80  -f  13s'  180  +  305  100 

where  the  multipliers  1  -f  —  r',  £c.  evidently  reduce  the  reading 

80 

to  what  it  would  have  been  if  the  observed  temperature  had  been 
that  of  freezing,  and  the  divisors  1  +  —  •  18,  £c.  further  reduce 

these  to  the  respective  temperatures  of  graduation,  and  conse- 
quently give  the  true  heights. 

This  true  height  of  the  mercury  will  be  proportional  to  the 
pressure  only  when  the  temperature'  of  the  mercury  is  constant. 
We  must,  therefore,  reduce  the  height  to  what  it  would  be  if  the 
temperature  were  equal  to  the  adopted  normal  temperature,  which 
is  in  our  table  8°  Reaumur  =  50°  F.  =  10°  C.  Now,  mercury 

expands  —  of  its  volume  at  the  freezing  point  of  water,  when 


SECOND    HYPOTHESIS. 


167 


its  temperature  if,  raised  from  that  point  to  the  boiling  point  of 
water.  Hence,  putting  q  =  — -,  the  above  heights  will  be  reduced 

to  the  normal  temperature  by  multiplying  them  respectively  by 
the  factors 


180  + 


80  -f  8g          

80  +  r'q          180  +  (/'  —  32)0' 


100  -f  IQg 
100  -{-c'q 


(225) 


The  normal  height  of  the  barometer  adopted  by  BESSEL  was  29.6 
inches  of  Bradley's  instrument,  or  333.28  Paris  lines ;  but  it  after- 
wards appeared  that  this  instrument  gave  the  heights  too  small 
by  J  a  Paris  line,  so  that  the  normal  height  in  the  tables  is  333.78 
Paris  lines,  at  the  adopted  normal  temperature  of  8°  R.  Reducing 
this  to  the  standard  temperature  of  the  Paris  line  =  13°  R.,  we 
have 


bn  =  333.78 


_0  80  +  8s 


80  -f-  135 


(226) 


In  comparing  this  with  the  observed  heights,  the  b(e)  and  6(m)  must 
be  reduced  to  lines  by  observing  that  one  English  inch  =  11.2595 
Paris  lines,  and  one  metre  =  443.296  Paris  lines.  Making  this 

reduction,  the  value  of  /9  —  -  is  found  by  dividing  the  product 

of  (224)  and  (225)  by  (226).  The  result  may  then  be  separated 
into  two  factors,  one  of  which  depends  upon  the  observed  height 
of  the  barometric  column,  and  the  other  upon  the  attached  ther- 
mometer ;  so  that  if  we  put 

B=      b(l)      80  +  8<? 
"333.78*80  -f  8s 


(e)    11.2595    80  4-  13s    180  -f  18g 
333.78      80  +  85  '  180  +  30s 


443.296   80  +  13s     100  -f  10  g 
'  "        100 


_    (m} 

'  333.78  '  80  4-  8s 


nid 


T^-. 


80  -f  r's  __  180  +  (/'  —  32)  s  __  100  4-  c's 
SO  -j-  r'2  ~  180  -|-  (/'  —  32)  q  ~  100  4-  c'q 


(227) 


\ve  shall 


or 


log  p  =  log  E  4-  log  T 


(228) 


168  REFRACTION. 

The  quantity  j-  would  be  computed  directly  under  the  form 


l+«(r-T.) 

if  r0  were  at  once  the  freezing  point  and  the  normal  temperature 
of  the  tables ;  for  s  is  properly  the  expansion  of  the  air  for  each 
degree  of  the  thermometer  above  the  freezing  point,  the  density 
of  the  air  at  this  point  being  taken  as  the  unit  of  density.  But 
if  the  normal  temperature  is  denoted  by  r0,  that  of  the  freezing 
point  by  rv  the  observed  by  r,  we  shall  have 


- 

" 


an  expression  which,  if  we  neglect  the  square  of  e,  will  be  reduced 
to  the  above  more  simple  one  by  dividing  the  numerator  and 
denominator  by  1  +  e(r0  —  r,).  BESSEL  adopted  for  r0  the  value 
50°  F.  by  BRADLEY'S  thermometer;  but  as  this  thermometer  was 
found  to  give  1°.25  too  much,  the  normal  value  of  the  tables  is 
TO  =  48°.  75  F.  Hence,  if  r,/,  or  c  denote  the  temperature  indi- 
cated by  the  external  thermometer,  according  as  it  is  Reaumur, 
Fahr.,  or  Cent.,  we  have* 

_  180  +  16.75  X  0.36438 


180  -f-  |  r  X  0.36438 
180  +  16.75  X  0.36438 


180  -f(/_32)  X  0.36438 

180  +  16.75  X  0.36438 
"l80~+~fc  X  0.36438 


(229) 


The  tables  constructed  according  to  these  formulae  give  the 
values  of  log  B,  log  JJ  and  log  ?-,  with  the  arguments  barometer, 
attached  thermometer,  and  external  thermometer  respectively 
and  the  computation  of  the  true  refraction  is  rendered  extreme' 
simple.     An  example  has  already  been  given  in  Art.  107. 

118.  In  the  preceding  discussion  we  have  omitted  any  con- 
sideration of  the   hygrometric   state  of  the  atmosphere.     The 

*  Tabulae  Regiomontanx,  p.  LXII. 


REFRACTION.  169 

refractive  power  of  aqueous  vapor  is  greater  than  that  of  at- 
mospheric air  of  the  same  density,  but  under  the  same  pressure 
its  density  is  less  than  that  of  air ;  and  LAPLACE  has  shown  that 
"  the  greater  refractive  power  of  vapor  is  in  a  great  degree  com- 
pensated by  its  diminished  density."* 

119.  Refraction  table  with  the  argument  true  zenith  distance. — When 
the  true  zenith  distance  f  is  given,  we  may  still  find  the  refrac- 
tion from  the  usual  tables,  or  Col.  A  of  Table  II.,  where  the 
apparent  zenith  distance  z  is  the  argument,  by  successive  ap- 
proximations. For,  entering  the  table  with  £  instead  of  z,  we 
shall  obtain  an  approximate  value  of  r,  which,  subtracted  from  £, 
will  give  an  approximate  value  of  z;  with  this  a  more  exact 
value  of  r  can  be  found,  and  a  second  value  of  z,  and  so  on,  until 
the  computed  values  of  r  and  z  exactly  satisfy  the  equation  z  = 
£  —  r.  But  it  is  more  convenient  to  obtain  the  refraction  directly 
with  the  argument  £.  For  this  purpose  Col.  B  of  Table  II.  gives 
the  quantities  a/,  A1 ',  //,  which  are  entirely  analogous  to  the  a,  A, 
and  A,  so  that  the  refraction  is  computed  under  the  form 

r  r=a'/?^Vx'  tan  C  (230) 

where  /9  and  7-  have  the  same  values  as  before. 

The  values  of  a',  A',  and  A'  are  deduced  from  those  of  a,  ^4, 
and  A  after  the  latter  have  been  tabulated.  They  are  to  be  so 
determined  as  to  satisfy  the  equations 

a,?^V  tan  2  =  a,'pA'Y*  tan  C  (231) 

*.«*  C  —  a' ;?'<>*' tan  C  (232) 

and  this  for  any  values  of  /3  and  f.  Let  (z)  denote  the  value  of  z 
which  corresponds  to  £  when  ft  =  1,  7  —  1 ;  that  is,  when  the 
refraction  is  at  its  mean  tabular  value.  The  value  of  (z)  may  be 
found  by  successive  approximations  from  Col.  A.,  as  above  ex- 
plained. Let  (a),  (A),  (/),  and  (r)  denote  the  corresponding 
values  of  a,  J.,  /\  r.  We  have 


(r)  =  (a)  tan  (<r)  =  a'  tan  C 

-    =  :  —  a'  tan 


whence,  by  (232), 


*  Mec.  Cel.  Book  X. 


170  REFRACTION. 

z  =  (2)  —  a'  tan  :  0?^>A'  —  1) 

But,  taking  Napierian  logarithms,  we  have 

l(p*r*)  =  A'ie  +  i'ir 
and  hence,  e  being  the  Napierian  base, 

pA'yV  _-  e  A'lft  +  \'ly  —  1    _J_   (4'  ip  _j_  X  Ifi   _f_  &C. 

where,  as  /9  and  f  differ  but  little  from  unity,  the  higher  powers 
of  A'lft  +  MY  may  be  omitted.     Hence 

2  =  (2)—  (r)  [A'l/9  +  Wr] 
Now,  taking  the  logarithm  of  (231),  we  have 

I  (»  tan  2)  4-  .4  J/9  +  llr  =  l  (a'  tan  C)  +  A'  1/3  -f  Wr 

The  first  member  is  a  function  of  2,  which  we  may  develop  as  a 
function  of  (z)  ;  for,  denoting  this  first  member  by  /e,  and  putting 


we  have  2  =  (2)  +  ?/,  and  hence 

+  y]  =/  ( 


where  we  may  also  neglect  the  higher  powers  of  y.  But  since 
f(z)  is  what/2  becomes  when  z  =  (2),  and  consequently  A  =  (^.), 
A  =  (^),  we  have 

/(*)  =  I  [(a)  tan  (2)]  +  (A)  ^  -f  (A)  /r 

tan    2  d0   tan   2  1      rfr 


d(2)   "  cZ  0)  (a)  tan  (2)  rf  (*)       (r) 

Hence  we  have 

fz  =  /  [(a)  tan  (5)]  -f  (A)  I  ft  +  (A)  Zr  -  [^  /p  4-  /  Jr] 

=  Z  [a'  tan  C]  +  A'  ^  -f  ;/  lr 
or,  since  (a)  tan  (2)  =  a'  tan  £, 


REFRACTION.  171 


Since  this  is  to  be  satisfied  for  indeterminate  values  of  /?  and  f, 
the  coefficients  of  //?  and  If  in  the  two  members  must  be  equal; 
and  therefore 

A'=- 


(233) 

and  also 

,      f  xtan(< 
a=Wta^ 

All  the  quantities  in  the  second  members  of  these  formulae  may 
be  found  from  Column  A  of  Table  II.,  and  thus  Column  B  may 
be  formed.* 
If  we  put 

we  shall  now  find  the  refraction  under  the  form 

r  =  k'  tan  C 

120.  To  find  the  refraction  of  a  star  in  right  ascension  and  decli- 
nation. 

The  declination  d  and  hour  angle  t  of  the  star  being  given, 
together  Avith  the  latitude  tp  of  the  place  of  observation,  we  first 
compute  the  true  zenith  distance  £  and  the  paral lactic  angle  q 
by  (20).  The  refraction  will  be  expressed  under  the  form 

r  =  k'  tan  C 
in  which 


The  latitude  and  azimuth  being  here  constant  (since  refrac- 
tion acts  only  in  the  vertical  circle),  we  have  from  (50),  by  put- 

*  See  also  BESSEL,  Astronomische  Unter$uchun<jen,  Vol.  I.  p.  159. 


172 


REFRACTION. 


ting  d(p  =  0,  dA  =  0,  d£  =  r  =  /j'  tan  £,  dt  =  —  da,  (a  =  star's 
right  ascension), 


d5  =  —  k'  tan  C  cos 
cos  oda  =  —  k'  tun  C  sin 


(234) 


which  are  readily  computed,  since  the  logarithms  of  tan  £  cos  q 
and  tan  £  sin  q  will  already  have  been  ftmnd  in  computing  £  by 
(20).  The  value  of  log  kr  will  be  found  from  Table  II.  Column 
B,  with  the  argument  £. 

The  values  of  dd  and  c/a  thus  found  are  those  which  are  to  be 
algebraically  added  to  the  apparent  declination  and  right  ascen- 
sion to  free  them  from  the  effect  of  refraction. 

The  mean  value  of  kr  is  about  57",  which  may  be  employed 
when  a  very  precise  result  is  not  required. 


Fig.  17. 


DIP    OF   THE    HORIZON. 

121.  The  dip  of  the  horizon  is  the  angle  of  depression  of  the 
visible  sea  horizon  below  the  true  horizon,  arising  from  the  ele- 
vation of  the  eye  of  the  observer  above  the  level  of  the  sea. 
Let  CZ,  Fig.  17,   be  the  vertical  line  of  an  observer  at  A, 

whose  height  above  the  level  of  the 
sea  is  AB.  The  plane  of  the  true  ho- 
rizon of  the  observer  at  A  is 'a  plane 
at  right  angles  to  the  vertical  line 
(Art.  3).  Let  a  vertical  plane  be 
passed  through  CZ,  and  let  ETD  be 
the  intersection  of  this  plane  with  the 
earth's  surface  regarded  as  a  sphere, 
AH  its  intersection  with  the  horizon- 
tal plane.  Draw  ATH(  in  this  plane, 
tangent  to  the  circular  section  of  the 
earth  at  T.  Disregarding  for  the  pre- 
sent the  effect  of  the  atmosphere,  T  will 
be  the  most  distant  point  of  the  surface  visible  from  A.  If  we 
now  conceive  the  vertical  plane  to  revolve  about  CZas  an  axis, 
AH  will  generate  the  plane  of  the  celestial  horizon,  while  All' 
will  generate  the  surface  of  a  cone  touching  the  earth  in  the 
small  circle  called  the  visible  horizon;  and  the  angle  HAH' 
will  be  the  dip  of  the  horizon. 


DIP    OF    THE    HORIZON. 


173 


122.    To  find  the  dip  of  the  horizon,  neglecting  the  atmospheric  refrac- 
tion.    Let 

x  —  the  height  of  the  eye  =  AB, 
a  =  the  radius  of  the  earth, 
D  =  the  dip  of  the  horizon. 

We  have  in  the  triangle  CAT,  ACT '=  HAH'  =  D,  and  hence 

AT 


tan  D  = 


CT 


By  geometry,  we  have 

AT  =  VAB  X  AD  =  Vx  (2 a  +  x) 
whence 


tan 


*=-    ^-V^NT? 


As  a?  is  always  very  small  compared  with  a,  the  square  of  the 


x  . 


fraction  --  is   altogether  inappreciable:  so  that  we   may  take 
simply 


tan  Z>  = 


(235) 


123.   To  find  the  dip  of  the  horizon,  having  regard  to  the  atmospheric 
refraction. 

The  curved  path  of  a  ray  of  light  from  the  point  T,  Fig.  18, 
to  the  eye  at  A,  is  the  same  as  that 
of  a  ray  from  A  to  T\  and  this  is 
a  portion  of  the  whole  path  of  a 
ray  (as  from  a  star  S)  which  passes 
through  the  point  A,  and  is  tangent 
to  the  earth's  surface  at  T.  ^The 
direction  in  which  the  observer  at 
A  sees  the  point  T  is  that  of  the 
tangent  to  the  curved  path  at  A,  or 
AH']  the  true  dip  is  therefore  the 
angle  HAH',  and  is  less  than  that  found  in  the  preceding  article. 
It  is  also  evident  that  the  most  distant  visible  point  of  the  earth's 


174  DIP    OF   THE    HORIZON. 

surface  is  more  remote  from  the  observer  than  it  would  be  if 
the  earth  had  no  atmosphere. 

Now,  recurring  to  the  investigation  of  the  refraction  in  Art. 
108,  we  observe  that  the  angle  HAH'  is  the  complement  of 
the  angle  of  incidence  of  the  ray  at  the  point  A,  there  denoted 
by  {;  and  it  was  there  shown  that  if  (/,  j«,  and  i  are  respectively 
the  normal,  the  index  of  refraction,  and  the  angle  of  incidence 
for  a  point  elevated  above  the  earth's  surface,  while  a,  /*0,  and  z 
are  the  same  quantities  at  the  surface,  we  have 

q  n  sin  i  =  a  /JEO  sin  z 
But  in  the  present  case  we  have  z  —  90°;  and  hence,  putting 

D'  =  the  true  dip  =  90°  —  i 
q    —  a  -|-  x 
we  have 


/*      a  -f  x         i 
Developing  and  neglecting  the  square  of  —  as  before, 


(236) 


which  would  suffice  to  determine  Df  when  //0  and  p.  have  been 
obtained  from  the  observed  densities  of  the  air  at  the  observer 
and  at  the  level  of  the  sea.  But,  as  D'  is  small,  it  is  more  con- 
venient to  determine  it  from  its  sine;  and  we  may  also  intro- 
duce the  density  of  the  air  directly  into  the  formula  by  putting 
(Art.  110), 


Substituting  the  value  of  a  from  (178),  namely, 

2H 


we  may  give  this  the  form 


DIP   OF   THE   HORIZON.  175 


which,  by  neglecting  the  square  of  the  second  term,  gives 


Hence,  still  neglecting  the  higher  powers  of  a  and  -,  as  well  as 
their  product,  we  have 

(237) 


which    agrees   with   the    formula    given    by    LAPLACE,    Mec    Gel 
Book  X. 

For  an  altitude  of  a  few  feet,  the  difference  of  pressure  will 
not  sensibly  affect  the  value  of  D',  and  may  be  disregarded, 
especially  since  a  very  precise  determination  of  the  dip  is  not 
possible  unless  we  know  the  density  of  the  air  at  the  risible  hori- 
zon, which  cannot  usually  be  observed.  We  may,  however, 
assume  the  temperature  of  the  water  to  be  that  of  the  lowest 
stratum  of  the  air,  and,  denoting  this  by  r0,  while  T  denotes  the 
temperature  of  the  air  at  the  height  of  the  eye,  we  have  [mak- 
ing p  =  p0  in  (171)],  approximately, 


in  which  for  Fahrenheit's  thermometer  e  =  0.002024.     Hence 


sin2  D 


where  D  is  the  dip,  computed  by  (235),  when  the  refraction  is 
neglected,  the  sine  of  so  small  an  angle  being  put  for  its  tan- 
gent. If  we  substitute  the  values  a  ==  0.00027895,  sin  D  = 
-D  sin  V,  and  e  =  0.002024,  this  formula  becomes 


176  DIP   OF   THE    HORIZON. 


in  which  D  is  in  seconds.     If  D  is  expressed  in  minutes  in  the 
last  term,  it  will  be  sufficiently  accurate  to  take 


(238) 


This  will  give  Df  =  D  when  r  =  r0,  as  it  should  do,  since  in 
that  case  the  atmosphere  is  supposed  to  he  of  uniform  density 
from  the  level  of  the  sea  to  the  height  of  the  observer.  If 
T  <  r0,  we  have  D'  >  D.  In  extreme  cases,  where  r  is  much 
greater  than  r0,  we  may  have  D'  <  0,  or  negative,  and  the  visible 
horizon  will  appear  above  the  level  of  the  eye,  a  phenomenon 
occasionally  observed.  I  know  of  no  observations  sufficiently 
precise  to  determine  whether  this  simple  formula,  deduced  from 
theoretical  considerations,  accurately  represents  the  observed 
dip  in  every  case. 

124.  If,  however,  we  wish  to  compute  the  value  of  Df  for  a 
mean  state  of  the  atmosphere  without  reference  to  the  actually 
observed  temperatures,  we  may  proceed  as  follows  :  In  the  equa- 
tion above  found, 


p.     a+x 

we  may  substitute  the  value 

In  \"+1_     a 
\;j.J  a  -\-x 

which  is  our  first  'hypothesis  as  to  the  law  of  decrease  of  density 
of  the  strata  of  the  atmosphere,  Art.  109.  This  hypothesis  will 
serve  our  present  purpose,  provided  n  is  so  determined  as  to 
represent  the  actually  observed  mean  horizontal  refraction.  We 
then, 


COB  IX  =(!+!)' 


and  developing,  neglecting  the  higher  powers  of—, 


DIP   OF   THE   HORIZON.  177 


n       1    a 


or 


To  determine  n,  we  have  by  (160),  reducing  r0  to  seconds, 

4H 


n  = 


(r0  sin  I")2 


where,  for  Barom.  Ow.76,  Therm.  10°C.,  which  nearly  represent 
the  mean  state  of  the  atmosphere  at  the  surface  of  the  earth,  we 
have  4k d0  =  0.00056795,  and  r0  =  34'  30"  (which  is  about  the 
mean  of  the  determinations  of  the  horizontal  refraction  by  dif> 
ferent  astronomers) ;  and  hence  we  find 


n  =  5.639,          /     n      =  0.9216  =  1  —  0.< 

\n  -f  1 


0.0784 


D'  =  D  —  .0784  D  (239) 

The  coefficient  .0784  agrees  very  nearly  with  DELAMBRE'S  value 
.07876,  which  was  derived  from  a  large  number  of  observations 
upon  the  terrestrial  refraction  at  different  seasons  of  the  year. 
To  compute  D'  directly,  we  have 


sin  1"   \  a 

If  x  is  in  feet,  we  must  take  a  in  feet.  Taking  the  mean  value 
a  =  20888625  feet,  and  reducing  the  constant  coefficient  of  i/x9 
we  have 


=  58".82  yx  in  feet.  (240) 


Table  XI.,  Vol.  II.,  is  computed  by  this  formula. 

VOL.  L— 12 


178  DISTANCE    OF   THE    HORIZON. 

125.   To  find  the  distance  of  the  sea  horizon^  and  the  distance  of  an 
object  of  known  height  just  visible  in  the  horizon. — The  small  portion 
TA,  Fig.  19,  of  the  curved  path  of  a  ray  of 
Fig.  19.  light,  may  be  regarded  as  the  arc  of  a  circle; 

and  then  the  refraction  elevates  A  as  seen 
from  T  as  much  as  it  elevates  T  as  seen 
from  A.  Drawing  the  tangent  TP,  the  ob- 
server at  T  would  see  the  point  A  at  P ; 
and  if  the  chord  TA  were  drawn,  the  angle 
PTA  would  be  the  refraction  of  A.  This 
refraction,  being  the  same  as  that  of  T  as 
seen  from  A,  is,  by  (239),  equal  to  .0784  Z).  In  the  triangle 
TPA,  TAP  is  so  nearly  a  right  angle  (with  the  small  elevations 
of  the  eye  here  considered)  that  if  we  put 

xl  =  AP 
we  may  take  as  a  sufficient  approximation 

xl  =  TA  X  tan  PTA  =  a  tan  D  X  -0784  tanD 
But  we  have  a  tan2jD  —  2x,  and  hence 

xl  =  .1568  x 
Putting 

d  —  the  distance  of  the  sea  horizon, 
we  have 


PT  =  y(£CB  -f  PB)  X  PB 
or,  nearly, 


d  =  i/2<7  (x  +  xj  =  1/2.3136  ax 

If  x  is  given  in  feet,  we  shall  find  d  in  statute  miles  by  dividing 
this  value  by  5280.  Taking  a  as  in  the  preceding  article,  we 
find 


l/2.3136a  _im 

5280 
and,  therefore, 

d  (in  statute  miles)  —  1.317  \/x  in  feet.  (241) 

If  an  observer  at  A'  at  the  height  A'Bf  =  xf  sees  the  object 
A,  whose  height  is  a:,  in  the  horizon,  he  must  be  in  the  curve  de- 


DIP   OF   THE   SEA.  179 

scribed  by  the  ray  from  A  which  touches  the  earth's  surface  at 
T.  The  distance  of  A'  from  Twill  be  =  1.317  |/P,  and  hence 
the  whole  distance  from  A  to  A'  will  be  =  1.317  (Vx  +  Vx'). 

The  above  is  a  rather  rough  approximation,  but  yet  quite  as 
accurate  as  the  nature  of  the  problem  requires  ;  for  the  anoma- 
lous variations  of  the  horizontal  refraction  produce  greater 
errors  than  those  resulting  from  the  formula.  By  means  of  this 
formula  the  navigator  approaching  the  land  may  take  advantage 
of  the  first  appearance  of  a  mountain  of  known  height,  to  deter- 
mine the  position  of  the  ship.  For  this  purpose  the  formula 
(241)  is  tabulated  with  the  argument  "  height  of  the  object  or 
eye  ;"  and  the  sum  of  the  two  distances  given  in  the  table,  cor- 
responding to  the  height  of  the  object  and  of  the  eye  respect- 
ively, is  the  required  distance  of  the  object  from  the  observer. 

126.  To  find  the  dip  of  the  sea  at  a  given  distance  from,  the  observer. 
—  By  the  dip  of  the  sea  is  here  understood  the  apparent  depres- 
sion of  any  point  of  the  surface  of  the  water  nearer  than  the 
visible  horizon.  Let  T,  Fig.  20,  be  such  a 
point,  and  A  the  position  of  the  observer. 
Let  TA'  be  a  ray  of  light  from  T,  tangent 
to  the  earth's  surface  at  TJ  meeting  the  ver- 
tical line  of  the  observer  in  A1  '.  Put 

D"  =  the  dip  of  T  as  seen  from  A, 

d    =  the  distance  of  T  in  statute  miles, 

x    =  the  height  of  the  observer's  eye  in  feet  =  AB, 

x'  =  A'B. 

We  have,  by  (241), 


and  the  dip  of  7]  as  seen  from  JL',  is,  therefore,  by  (240), 
=  58".82  /p  =  44".66  d. 


Now,  supposing  the  chords  TA,  TA'  to  be  drawn,  the  dip  of  T 
at  A  exceeds  that  at  A'  by  the  angle  A  TA',  very  nearly ;  and 
we  have  nearly 

<*•]  >  A  TA' ^^  ^  x  —  x' 

~  -pTj-f  X  sin  in  —  5280  d  sin  I" 


180  SEMIDIAMETERS. 

whence 

&'  =  44".66  d+ 


5280  d  sin  V 
Substituting  the  value  of  x'  in  terms  of  d, 

D"  =  22".  U  d  -f-  39".07  -  (x  being  in  feet  and  d  in  statute 

d 
miles).  (242) 

If  f/  is  given  in  sea  miles,  we  find,  by  exchanging  d  for  —^d, 

7)"  =  25".65  d  -f  33".73  —(a;  being  in  feet  and  d  in  sea 

d 
miles).  (243) 

The  value  of  D"  is  given  in  nautical  works  in  a  small  table 
with  the  arguments  x  and  d.  The  formula  (243)  is  very  nearly 
the  same  as  that  adopted  by  BOWDITCH  in  the  Practiced  Navigator. 

127.  At  sea  the  altitude  of  a  star  is  obtained  by  measuring  its 
angular  distance  above  the  visible  horizon,  which  generally 
appears  as  a  well-defined  line.  The  observed  altitude  then 
exceeds  the  apparent  altitude  by  the  dip,  remembering  that  by 
apparent  altitude  we  mean  the  altitude  referred  to  the  true 
horizon,  or  the  complement  of  the  apparent  zenith  distance. 
Thus,  h'  being  the  observed  altitude,  h  the  apparent  altitude, 


or,  when  the  star  has  been  referred  to  a  point  nearer  than  the 
visible  horizon, 

h  =  h'  —  D" 

SEMIDIAMETERS  OF  CELESTIAL  BODIES. 

128.  In  order  to  obtain  by  observation  the  position  of  the 
centre  of  a  celestial  body  which  has  a  well-defined  disc,  we 
observe  the  position  of  some  point  of  the  limb  and  deduce  that 
of  the  centre  by  a  suitable  application  of  the  angular  semi- 
diameter  of  the  body. 

I  shall  here  consider  only  the  case  of  a  spherical  body.  The 
apparent  outline  of  a  planet,  whether  spherical  or  spheroidal, 
and  whether  fully  or  partially  illuminated  by  the  sun,  will  be 


SEMIDIAMETERS. 


181 


Fig.  21. 


discussed   in   connection   with    the    theory   of    occupations   in 
Chapter  X. 

The  angular  semidiaraeter  of  a  spherical  body  is  the  angle 
subtended  at  the  place  of  observation  by  the  radius  of  the  disc. 
I  shall  here  call  it  simply  the  semidiameter,  and  distinguish  the 
linear  semidiameter  as  the  radius. 

Let  0,  Fig.  21,  be  the  centre  of 
the  earth,  A  the  position  of  an  ob- 
server on  its  surface,  M  the  centre 
of  the  observed  body;  OB,  AB', 
tangents  to  its  surface,  drawn  from 
0  and  A.  The  triangle  OBM  re- 
volved about  0  M  as  an  axis  will  de- 
scribe a  cone  touching  the  spherical 
body  in  the  small  circle  described 
by  the  point  B,  and  this  circle  is  the 
disc  whose  angular  semidiameter  at 
0  is  M  OB.  Put 

S  =  the  geocentric  semidiametcr,  MOB, 
S'  =  the  apparent  scraidiamcter,  MAB', 
A,  J'  =  the  distances  of  the  centre  of  the  body  from  the  centre  of 

the  earth  and  the  place  of  observation  respectively, 
a  =  the  equatorial  radius  of  the  earth, 
a'  —  the  radius  of  the  body, 


then  the  right  triangles  OMB,  A  MB  '  give 


sin  S  = 


a' 
T7 


(244) 


But  if 

TT  —  the  equatorial  horizontal  parallax  of  the  body, 

we  have,  Art.  89, 


and  hence 


sin  ic  =  — 
A 


sin  S  =  —  sin 
a 


sin  Sr  =  —  sin  S 
A' 


or,  with  sufficient  precision  in  most  cases, 


c       a 

S  =  — 


(245) 


(246) 


182  SEMIDIAMETERS. 

The  geocentric  semidiameter  and  the  horizontal  parallax  have 
therefore  a  constant  ratio  =  —  .  For  the  moon,  we  have 

-  ==  0.272956  (247) 

a 

as  derived  from  the  Greenwich  observations  and  adopted  by 
HANSEN  (Tables  de  la  Lune,  p.  39). 

If  the  body  is  in  the  horizon  of  the  observer,  its  distance  from 
him  is  nearly  the  same  as  from  the  centre  of  the  earth,  and  hence 
the  geocentric  is  frequently  called  the  horizontal  semidiameter; 
but  this  designation  is  not  exact,  as  the  latter  is  somewhat  greater 
than  the  former.  In  the  case  of  the  moon  the  difference  is 
between  0".l  and  0".2.  See  Table  XII. 

If  the  body  is  in  the  zenith,  its  distance  from  the  observer  is 
less  than  its  geocentric  distance  by  a  radius  of  the  earth,  and  the 
apparent  semidiameter  has  then  its  greatest  value. 

The  apparent  semidiameter  at  a  given  place  on  the  earth's 
surface  is  computed  by  the  second  equation  of  (245)  or  (246),  in 

which  the  value  of  —  is  that  found  by  (104)  ;  so  that,  putting  z  = 

the  true  (geocentric)  zenith  distance  of  the  body,  £'  —  the  appa- 
rent zenith  distance  (affected  by  parallax),  A  --  its  azimuth, 
<p  —  <f>f  the  reduction  of  the  latitude,  we  have,  (by  (111)  and  (104), 


sin  (C  —  r) 

129.  This  last  formula  is  rigorous,  but  an  approximate  formula 
for  computing  the  difference  S1  —  $will  sometimes  be  convenient. 
In  (103)  we  may  put 

cos       —    ' 


COS  f  COS  I  (C'  —  C) 

without  sensible  error  in  computing  the  very  small  difference  in 
question  ;  we  thus  obtain 

A' 

-  =  1  —  p  sin  TT  cos  [-J  (C'  -}-  C)  —  f] 
A 


SEMIDIAMETERS.  183 

Putting 

m  =  p  sin  rr  cos  [*  (C'  +  C)  —  r]  (249) 

we  have 

—  = =  1  -f  m  -f  w2  -f  &c. 

J'       1-ro 

s 

and  hence,  since  the  third  power  of  m  is  evidently  insensible, 

S'  _  s  =  Sw  +  Sm2  (250) 

which  is  practically  as  exact  as  (248).     The  value  of  £'  required 
in  (249)  will  be  found  with  sufficient  accuracy  by  (114),  or 

C'  —  C  =  p  *  sin  (;'  —  r) 

The  quantity  Sf  —  S  is  usually  called  the  augmentation  of  the 
semidiameter.  It  is  appreciable  only  in  the  case  of  the  moon. 

130.  If  we  neglect  the  compression  of  the  earth,  which  will 
not  involve  an  error  of  more  than  0".05  even  for  the  moon,*  we 
may  develop  (250)  as  follows.  Putting  p  =  1  and  f  =  0  in  (249), 
we  may  take 

m  =  sin  TT  cos  }  (£'  +  C) 

=  sin  *  cos  [:'  —  *  (:'  — :)] 

=  sin  -K  cos  Z'  -\-  \  sin  r  sin  (C'  —  C)  sin  C 
=  sin  TT  cos  C'  -f-  J  sin2  JT  sin2  C' 

which  substituted  in  (250)  gives,  by  neglecting  powers  of  sin  x 
above  the  second, 

S'  —  S  =  S  sin  TT  cos  C' -f  IS  sin2*  sin2C'-f  S  sin2*  cos2C' 
=  S  sin  *  cos  C'  +  *  £  sin2*  -f  *  £  sin2*  cos2  C' 


But  we  have 


a'          a'     sin 

O  =•    TT  =  • 


«     sin  1" 


*  The  greatest  declination  of  the  moon  being  less  than  30°,  it  can  reach  great 
altitudes  only  in  low  latitudes,  where  the  compression  is  less  sensible.  A  rigorous 
investigation  of  the  error  produced  by  neglecting  the  compression  shows  that  the 
maximum  error  is  less  than  0".06. 


184  SEMIDIAMETERS. 

arid  if  we  put 

h  =  2-r  sin  1",  log  h  =  5.2495 

we  have  sin  TT  =  h  S,  which  substituted  above  gives  the  follow- 
ing formula  for  computing  the  augmentation  of  the  moon's 
semidiameter  : 

S'  —  S  =  h  S2  cos  C'  +  £  A2  S*  +  £  A2  £3  cos2  C'  (251) 

EXAMPLE.  —  Find  the  augmentation  for  £  =  40°,  S  =  16'  0" 
—  960". 

log  £2  5.9645  log  S3  8.947  1st  term  =  12".54 

log  h  5.2495  log  £  /i2         0.198  2d       "    =    0  .14 

log  cos  C'       9.8843  log  2d  term  9.145  3d       «    =    0.  08 

log  1st  term  1.0983  log  cos2  C'     9.769  S'  —  S  =  12  .76 

log  3d  term  8.914 

The  value  of  £'  —  S  may  be  taken  directly  from  Table  XII.  with 
the  argument  apparent  altitude  =  90°  —  £'. 

131.  If  the  geocentric  hour  angle  (t)  and  declination  (3)  are 
given,  we  have,  by  substituting  (137)  in  (245), 


. 

sin  (o  — 


(252) 


for  which  7-  and  df  are  to  be  determined  by  (134)  and  (136),  or 
with  sufficient  accuracy  for  the  present  purpose  by  the  formulae 

tan  <pf 
tan  f=. 


cos  t 


dr  —  d  —  p  *  Sin 


sn 


132.  To  find  the  contraction  of  the  vertical  semidiameter  of  the  sun 
or  moon  produced  by  atmospheric  refraction. 

Since  the  refraction  increases  with  the  zenith  distance,  the 
refraction  for  the  centre  of  the  sun  or  the  moon  will  be  greater 
•than  that  for  the  upper  limb,  and  that  for  the  lower  limb  will  be 
greater  than  that  for  the  centre.  The  apparent  distance  of  the 


SEMIDIAMETERS.  185 

limbs  is  therefore  diminished,  and  the  whole  disc,  instead  of 
being  circular,  presents  an  oval  figure,  the  vertical  diameter  of 
which  is  the  least,  and  the  horizontal  diameter  the  greatest. 
The  refraction  increasing  more  and  more  rapidly  as  the  zenith 
distance  increases,  the  lower  half  of  the  disc  is  somewhat  moro 
contracted  than  the  upper  half. 

The  contraction  of  the  vertical  semidiameter  may  be  found 
directly  from  the  refraction  table,  by  taking  the  difference  of 
the  refractions  for  the  centre  and  the  limb. 

EXAMPLE. — The  true  semidiameter  of  the  moon  being  16'  0", 
and  the  apparent  zenith  distance  of  the  centre  84°,  find  the  con- 
traction of  the  upper  and  lower  semidiameters  in  a  mean  state 
of  the  atmosphere  (Barom.  30  inches,  Therm.  50°  F.).  We  find 
from  Table  I. 

For  apparent  zcn.  dist.  of  centre,      84°    0'  Refr.  =  8'  28".0 

"    approx.          "         upper  limb,  83    44  "=89  .4 

"         "  "         lower     "      84    16  «      =8  48  .1 

Hence, 

Approx.  contraction  tipper  semid.  =  8'  28".0  —  8'    9".4  =  18".6 
«  "  lower      «       =  8  48  .1  —  8  28  .0  =  20  .1 

These  results  are  but  approximate,  since  we  have  supposed  the 
apparent  zenith  distance  of  the  limb  to  differ  from  that  of  the 
centre  by  the  true  semidiameter,  whereas  they  differ  only  by  the 
apparent  or  contracted  semidiameter.  Hence  we  must  repeat  as 
follows: 

App.  zen.  dist.  upper  limb  =  83°  44'  18".G  Refr.  =  8'    9".7 

"          "         lower     «     =  84    15  39  .9  "      =  8  47  .7 

Contraction  of  upper  semid.  =  8'  28".0  —  8'    9".7  =  18".3 
"  lower      «      =  8  47  .7  —  8  28  .0  =  19  .7 

Observations  at  great  zenith  distances,  where  this- contraction 
is  most  sensible,  do  not  usually  admit  of  great  precision,  on 
account  of  the  imperfect  definition  of  the  limbs  and  the  uncer- 
tainty of  the  refraction  itself.  It  is,  therefore,  sufficiently  exact 
to  assume  the  contraction  of  either  the  upper  or  lower  semi- 
diameter  to  be  equal  to  the  mean  of  the  two.  In  the  above 
example,  which  offers  an  extreme  case,  if  we  take  the  mean 


186  SEMIDIAMETERS. 

19"  0  as  the  contraction  for  either  semidiameter,  the  error  will 
be  only  0".7,  which  is  quite  within  the  limit  of  error  of  observa- 
tions at  such  zenith  distances. 

133.   To  find  the  contraction  of  any  inclined  semidiameter -,  produced 

by  refraction. 

Let  M,  Fig.  22,  be  the  apparent  place  of  the  sun's  or  the 
moon's  centre;  ACBD,  a  circle  described 
with  a  radius  MA  equal  to  the  true  semi- 
diameter,  will  represent  the  disc  as  it  would 
appear  if  the  refraction  were  the  same  at 
all  points  of  the  limb.  The  point  A,  how- 
D  ever,  being  less  refracted  than  M,  will  ap- 
pear at  A',  P  at  P',  &c. ;  while  B,  being 
more  refracted  than  M,  appears  at  Bf.  The 
contraction  is  sensible  only  at  great  zenith 
distances,  where  we  may  assume  that  AM 

and  PP'E,  small  portions  of  vertical  circles  drawn  through  A 

and  P,  ar^  sensibly  parallel.     If  then  we  put 

S  =  the  true  vertical  semidiameter  =  AM, 
S^—  the  contracted  vert,  semid.  =  A'M, 
$a—  the  contracted  inclined  semid.  =  MPf,  which  makes  an 

angle  q  with  the  vertical  circle, 

A^=  the  contraction  of  the  vertical  semid.  =  8  —  St 
A$e  =  the  contraction  of  tho  inclined  semid.  =  S —  S9 

we  shall  have 

&9  cos  q  =  P'E  =  the  difference  of  the  apparent  zenith  distances 
Sl  =  tho  difference  of  the  app.  zen.  dist.  of  M  and  A'. 

Now,  the  difference  of  the  refractions  at  M  and  A'  is  AA',  and 
the  difference  of  the  refractions  at  M  and  P'  is  PPr ;  and,  since 
these  small  differences  are  nearly  proportional  to  the  differences 
of  zenith  distance,  we  have 

S, :  Sq  cos  q  =  AA' :  PP' 


SEMIDIAMETERS.  187 

The  small  triangle  PFP'  may  be  regarded  as  rectilinear  and 
right-angled  at  F;  whence 

FP'  =  PP'  X  cos  q 
or 


If  we  put  Sl  for  Sq  in  the  second  member,  the  resulting  value  of 
ASq  will  never  be  in  error  0".2  for  zenith  distances  less  than  85°, 
and  it  suffices  to  take 

J£,=:  J^cos2?  (253) 

This  formula  is  sufficiently  exact  for  all  purposes  to  which  we 
shall  have  occasion  to  apply  it. 

134.   To  find  the  contraction  of  the  horizontal  semidiameter.  —  The 
formula  (253)  for  q  =  90°  makes  the  contraction  of  the  hori- 
zontal semidiameter  =  0.     This  results  from  our  having  assumed 
that  the  portions  of  vertical  circles  drawn  through  the  several 
points  of  the  limb  are  parallel,  and  this  assumption  de- 
parts most  from  the  truth  in  the  case  of  the  two  ver- 
tical  circles   drawn   through    the    extremities   of  the 
horizontal  diameter.     To  investigate  the  error  in  this 
case,  let  ZM,  Fig.  23,   be   the  vertical    circle   drawn 
through   the    centre    of   the   body,    ZM  '  that   drawn 
through  the  extremity  of  the  horizontal  semidiameter 
MM'.     In  consequence  of  the  refraction,  the  points  M 
and  Mf  appear  at  JVand  JV',     If  we  denote  the  zenith 
distances  of  M  and  N  by  £  and  z,  those  of  Mf  and  Nf 
by  *   and  2',  the  refraction  MN  may  be  expressed  as  a  func- 
tion cither  of  z  or  of  f  ,  Art.  107,  and  we  shall  have 

r  =  A  tan  z-  =  k'  tan  C 

where  k  and  kr  are  given  by  the  refraction  table  with  the  argu- 
ments z  and  £.  The  zenith  distance  of  the  point  M'  differs  so 
little  from  that  of  M  that  the  values  of  k  and  k'  will  be  sensibly 
the  same  for  both  points,  and  we  shall  have  for  the  refraction 
M'N', 

r'  =  k  tan  2'  =  A-'  tan  C' 


188  SEMIDIAMETERS. 

These  two  equations  give 

tan  z        tan  C 


tan  *'       tan  C' 
But  if  the  triangle  ZNN'  is  right-angled  at  N,  we  have 

„      tan  z 

cos  Z  = • 

tan*?' 

and  hence,  also, 

tan  C 

cos  Z  = 

tan;' 

TJierefore  the  triangle  ZMM'  is  also  right-angled,  and  it  gives 

tan  S          tan  S' 

tan  Z  =  -7—         -  =  ^-r- 
sin  (2  -}-  r)         sin  2 

in  which  5  —  MM'  and  /S"  —  J!W.     Hence 

tan  S       sin  (£  4-  r) 

-~7  =  -  --. =  cos  r  -4-  sin  r  cot  z 

tan  fir  sin  2 

or,  very  nearly, 

£   =  1  +  r  sin  1"  cot  *  =  1  +  A  sin  1" 

o 

Hence  the  contraction  of  the  horizontal  semidiameter  is  ex- 
pressed by  the  following  formula : 

S—  S'=S'kamI" 

In  the  zenith,  the  mean  value  of  log  k  is  1.76156;  at  the  zenith 
distance  85°, it  is  1.71020.  For  Sr  =  16',  therefore,  the  contrac- 
tion found  by  this  formula  is  0".27  in  the  zenith,  and  0".24  for 
85°.  Thus,  for  all  zenith  distances  less,  than  85°  the  contraction  of 
the  horizontal  semidiameter  is  very  nearly  constant  and  equal  to  one- 
fourth  of  a  second. 

When  the  body  is  in  the  horizon,  we  have  k  =  rcot  2  =  0, 
and  hence  S  —  S'  =  0,  which  follows  also  from  the  sensible 
parallelism  of  the  vertical  circles  at  the  horizon. 


REDUCTION    OF    ZENITH    DISTANCES.  189 


REDUCTION    OF    OBSERVED    ZENITH     DISTANCES    TO    THE     CENTRE    OF 

THE    EARTH. 

135.  It  is  important  to  observe  a  proper  order  in  the  applica- 
tion of  the  several  corrections  which  have  been  treated  of  in  this 
chapter. 

The  zenith  distance  of  any  point  of  the  heavens  observed  with 
any  instrument  is  generally  affected  with  the  index  error  and 
other  instrumental  errors.  These  errors  will  be  treated  of  in 
the  second  volume  ;  here  we  assume  that  they  have  been  duly 
allowed  for,  and  we  shall  call  "observed"  zenith  distance  that 
which  would  be  obtained  with  a  perfect  instrument,  and  shall 
denote  it  by  z. 

In  all  cases  the  first  step  in  the  reduction  is  to  find  the  refrac- 
tion r  (—  a^fHan  z)  with  the  argument  £,  and  then  z  +  r  is  tho 
zenith  distance  freed  from  refraction. 

1st.  In  the  case  of  &  fixed  star, 


is  at  once  the  required  geocentric  zen.  dist. 

2d.  In  the  case  of  the  moon,  the  zenith  distance  observed  is 
that  of  the  upper  or  lower  limb.  If  S  is  the  geocentric  and  S' 
the  augmented  semidiameter  found  by  Art.  128,  129,  or  130, 

C'  =  z  -f  r  ±  S' 

is  the  apparent  zenith  distance  of  the  moon's  centre  freed  from 
refraction,  and  affected  only  by  parallax,  and,  consequently,  it  is 
that  which  has  been  denoted  by  the  same  symbol  in  the  discus- 
sion of  the  parallax.  "With  this,  therefore,  we  compute  the 
parallax  in  zenith  distance,  £'  —  f  ,  by  Art.  95,  and  then 


is  the  required  geocentric  zenith  distance  of  the  moon's  centre. 

To  compute  S'  by  (248),  (250),  or  (251),  we  must  first  know  £'; 
but  it  will  suffice  to  employ  in  these  formulae  the  approximate 
value  £  =  z  +  r  ±  & 

We  can,  however,  avoid  the  computation  of  #',  when  extreme 
precision  is  not  required,  by  computing  the  parallax  for  the 
zenith  distance  of  tho  limb.  Thus,  putting  £'  =  z  -f-  r,  and 


190  REDUCTION    OF    ZENITH    DISTANCES. 

computing  £'  —  £  by  Art,  95,  the  quantity  £  =  C'  —  (C'  —  C)  is 
the  geocentric  zenith  distance  of  the  limb;  and  therefore,  ap- 
plying the  geocentric  semidiameter,  £  ±  S  is  the  required  geo- 
centric zenith  distance  of  the  moon's  centre.  This  process 
involves  the  error  of  assuming  the  horizontal  parallax  for  the 
limb  to  be  the  same  as  that  for  the  moon's  centre.  It  can  easily 
be  shown,  however,  that  the  error  in  the  result  will  never  amount 
to  0".2,  which  in  most  cases  in  practice  is  unimportant.  The 
exact  amount  will  be  investigated  in  the  next  article. 

3d.  In  the  case  of  the  sun  or  a  planet,  when  the  limb  has  been 
observed,  the  process  of  reduction  is,  theoretically,  the  same  as 
for  the  moon  ;  but  the  parallax  is  so  small  that  the  augmentation 
of  the  semidiameter  is  insensible.  We  therefore  take 

C  =  z  +  r  ±S 

and  then,  computing  the  parallax  by  Art.  96,  or  even  by  Art.  90, 
£  =  £'  —  (£'  —  £)  is  the  true  geocentric  zenith  distance. 

If  a  point  has  been  referred  to  the  sea  horizon  and  the 
measured  altitude  is  H,  then,  D  being  the  dip  of  the  horizon, 
hf  =  H—  D  is  properly  the  observed  altitude,  and  z  —  90°  —  h' 
the  observed  zenith  distance,  with  which  we  proceed  as  above. 

136.  The  process  above  given  for  reducing  the  observed  zenith 
distance  of  the  moon's  limb  to  the  geocentric  zenith  distance  of 
the  moon's  centre,  is  that  which  is  usually  employed ;  but  the 
whole  reduction,  exclusive  of  refraction,  may  be  directly  and 
rigorously  computed  as  follows.  Putting 

C'  =  z  -f  r  =  the  apparent  zenith  distance  of  the  moon's  limb 

corrected  for  refraction, 
C  =  the  geocentric  zenith  distance  of  the  moon's  centre, 

then,  S'  being  the  augmented  semidiameter,  we  must  substitute 
£'  ±  S'  for  £  in  the  formulae  for  parallax,  and,  by  (101),  we 
have 

/  sin  (C'  ±  £')  —  sin  C  —  f>  sin  n  cos  (?>  —  ?>')  tan  ? 
f  cos  (C'  ±  S')  =  cos  C  —  p  sin  TT  cos  (y  —  <?') 

Multiplying  the  first  of  these  by  cos  £',  the  second  by  sin  £',  and 
subtracting,  we  have 


REDUCTION    OF    ZENITH    DISTANCES.  191 

±/sin  8'  =  -  sin  (C'  -  C)  +  /"'".'«*(?-/)  sin  (f,         ) 

COS  7- 

Af 

in  which /=  -.     By  (245)  we  have  also 

/  sin  S'  =  sin  £ 
and  hence  the  rigorous  formula 

sin  (:'  -  C)  =  P  sin  *  sin  (:'  -  r}  cos  (?  ~  ?0  -  sin  8 

COS  f 

for  which,  however,  we   may  employ  with   equal  accuracy  in 
practice 

sin  (C'  —  C)  =  P  sin  r.  sin  (:'  —  r}  q:  sin  S  (254) 

in  which,  J.  being  the  moon's  azimuth,  we  have 

r  =  (</>  —  <?')  cos  A  ...::,.: .:: 

If  we  put  (Art.  128) 

k  =  -  =  0.272956 
a 

we  have  sin  S=k  sin  TT,  and  (254)  may  be  written  as  follows : 

sin  (:'  —  C)  =  [>  sin  (:'  —  r)  =p  A']  sin  ?r  (255) 

For  convenience  in  computation,  however,  it  will  be  better  to 
make  the  following  transformation.     Put 

sin p  =  p  sin  r.  sin  (C'  —  r)  (256) 

then  (254)  becomes 

sin  (C' —  C)  =  sinp  =p  sin  S 

=  sin  (p  qp  £)  -|-  sin  p  (1  —  cos  $)  =p  sin  /Sf  (1  —  COB  p) 
=  sin  (|>  nF  <S)  H-  2  sin  p  sin2  £  /S  q=  2  sin  £  sin2  J  p 

where  the  last  two  terms  never  amount  to  0".2,  and  therefore  the 
formula  may  be  considered  exact  under  the  form 

sin  (C'  -  -  C)  =  sin  (p  qi  S)  =P  i  (^  nF  #)  sin  1"  sin  ^  sin  S 
Since  £'  —  f  and  p  +  S  differ  by  so  small  a  quantity,  there  will 


192  REDUCTION   OF   ZENITH   DISTANCES. 

be  no  appreciable  error  in  regarding  them  as  proportional  to 
their  sines ;  and  hence  we  have 

C'  —  :  =  p  =F  S  +  £  (p  =F  S)  sin  p  sin  S  (257) 

the  upper  signs  being  used  for  the  upper  limb  and  the  lower 
signs  for  the  lower  limb. 

In  this  formula,  p  is  the  parallax  computed  for  the  zenith 
distance  of  the  limb,  and  the  small  term  %(p  =£  S)sm  p  sin  $  may 
be  regarded  as  the  correction  for  the  error  of  assuming  the 
parallax  of  the  limb  to  be  the  same  as  that  of  the  centre. 

EXAMPLE.— In  latitude  <p  —  38°  59'  K,  given  the  observed  zenith 
distance  of  the  moon's  lower  limb,  z  =  47°  29'  58",  the  azimuth 
A  =  33°  0',  Barom.  30.25  inches,  At.  Therm.  65°  F.,  Ext.  Therm. 
64°  F.,  Eq.  hor.  par.  x  =  59'  10".20 ;  find  the  geocentric  zenith 
distance  of  the  moon's  centre  : 

(Table  III.)           fo  —  f»)  =  11'  15"  *  =  47°  29'58".00 

log  (?  -  ?')  =  2.8293                         (Table  II.)   r  =_       1    2.27 

log  cos  A  9.9236  C'  =  47  31    0  .27 

log  y  2.7529  7  =          9  26  • 

(Table  III.)  log  p  9.999428  C'  —  Y  —  *~  21  34  . 

log  sin  TT  .       8.235806 

log  sin  (C'  —  Y)  9.866652 

log  sin  p  8.101886  p  =  43'  28".09 

log  sin  TT  8.235806  S  =  16     9.00 

(Art.  128)  log  (0.272956)  9.436093  p  +  S  =  59  37  .09 

log  sin  S  7.071899         £  (p  -f-  S)  sin  p  sin  S  =  0.11 

log  sin  jo  sin  S  5.7739  C'  —  C  =  59  37  .20 

log  (p  -f  S)  3.5535 

log  I  9.6990  £  =  46°  31' 23".  07 

log  £  (p  -f  S)  siup  sin  S  9.0264 

It  is  hardly  necessary  to  observe  that  if  the  geocentric  zenith 
distance  of  the  centre  of  the  moon  or  other  body  is  given,  the 
apparent  zenith  distance  of  the  limb  affected  by  parallax  and 
refraction  will  be  deduced  by  reversing  the  order  of  the  steps 
above  explained. 

If  altitudes  are  given,  we  may  employ  altitudes  throughout 
the  computation,  putting  everywhere  90°  —  /?,  &c.  for  z,  &c.,  and 
making  the  necessary  obvious  modifications  in  the  formulae. 


TIME    BY    OBSERVATIONS.  193 


CHAPTER    V. 

FINDING   THE   TIME   BY   ASTRONOMICAL   OBSERVATIONS. 

137.  WE  have  seen,  Art.  55,  that  the  local  time  at  any  place 
is  readily  found  when  the  hour  angle  of  any  known  heavenly 
body  is  given.  This  hour  angle  is  obtained  by  observation,  but, 
a  direct  measure  of  it  being  in  general  impracticable,  we  must 
have  recourse  tp  observations  from  which  it  can  be  deduced. 

The  observer  is  supposed  to  be  provided  with  a  clock,  chro- 
nometer, or  watch,  which  is  required  to  show  the  time,  mean  or 
sidereal,  either  at  his  own  or  at  some  assumed  meridian,  such  as 
that  of  Greenwich. 

The  clock  correction*  is  the  quantity  which  must  be  added  alge- 
braically to  the  time  shown  by  the  clock  to  obtain  the  correct 
time  at  the  meridian  for  which  the  clock  is  regulated.  If  we  put 

T  =  the  clock  time, 
T'  —  the  true  time, 
AT  =  the  clock  correction, 
we  have 

T'  =  T  -f-  AT 
or  t  *T  =  T'—T  (258) 


and  the  clock  correction  will  be  positive  or  negative,  according  as 
the  clock  is  slow  or  fast.  It  is  generally  the  immediate  object  of 
an  observation  for  time  to  determine  this  correction.  At  the 
instant  of  the  observation,  the  time  T  is  noted  by  the  clock, 
and  if  this  time  agrees  with  the  time  T'  computed  from  the 
observation,  the  clock  is  correct;  otherwise  the  clock  is  in  error, 
and  its  correction  is  found  by  the  equation  &T=  T'  --  T. 

The  clock  rate  is  the  daily  or  hourly  increase  of  the  clock  cor- 
rection.    Thus,  if 

*  For  brevity,  I  shall  use  clock  to  denote  any  time-keeper. 

VOL.  I.—  13 

0 


194  TIME    BY    OBSERVATIONS. 

A  7^  =  the  clock  correction  at  a  time  T9, 
AT  =  "  "  "      T, 

dT  =  the  clock  rate  in  a  unit  of  time, 
we  have 

±T=  AT;  +  8T  (T  —  T0)  (259) 

where  T --  T0  must  be  expressed  in  days,  hours,  &c.,  according 
as  3Tis  the  rate  in  one  day,  one  hour,  &c. 

When,  therefore,  the  clock  correction  and  rate  have  been 
found  at  a  certain  instant  TQ,  we  can  deduce  the  true  time  from 
the  clock  indication  T  (or  "clock  face,"  as  it  is  often  called) 
at  any  other  instant,  by  the  equation 

T'  =  T+  Aro-f  ST(T—  TQ)  (260) 

If  the  clock  correction .  has  been  determined  at  two  different 
times  TQ  and  T,  the  rate  is  inferred  by  the  equation 


But  these  equations  are  to  be  used  only  so  long  as  we  can 
regard  the  rate  as  constant. 

Since  such  uniformity  of  rate  cannot  be  assumed  for  any  great 
length  of  time,  even  with  the  best  clocks  (although  the  perform- 
ance of  some  of  them  is  really  surprising),  it  is  proper  to  make 
the  interval  between  the  observations  for  time  so  small  that  the 
rate  may  be  taken  as  constant  for  that  interval.  The  length  of 
the  interval  will  depend  upon  the  character  of  the  clock  and  the 
degree  of  accuracy  required. 

EXAMPLE. — At  noon,  May  5,  the  correction  of  a  mean  time 
clock  is  —  16"1  47*,30  ;  at  noon,  May  12,  it  is  —  16W  13*.50 ;  what  is 
the  mean  time  on  May  25,  when  the  clock  face  is  11A  13™  12*.6, 
supposing  the  rate  to  be  uniform  ? 

May    5,  corr.    —  —  16W  47'.30 
«     12,     "        =  —  16    13.50 
Eate  in  7  days  —       -j-    33  .80 
3T=       +      4.829 

Taking,  then,  as  our  starting  point  TQ  =  May  12,  0^,  we  have 


TIME.  195 

for  the  interval  to  T=  May  25,  11*  13™  12-.6,  T—T0  =  13*  11* 
13m  12S.6  =  13^467.     Hence  we  have 


A  T0  =  —  16"13'.50 

dT(T-  T0)  =  -f     1      5  .03 

*T=—  15      8.47 

T  =  11M3»12«.6Q 

T'  =  10  58     4JL3 


But  in  this  example  the  rate  is  obtained  for  one  true  mean 
day,  while  the  unit  of  the  interval  13d.467  is  a  mean  day  as 
shown  by  the  clock.  The  proper  interval  with  which  to  com- 
pute the  rate  in  this  case  is  13d  10*  58"*  4M3  =  13rf.457  with 
which  we  find 


=  —    16M3-.50 
dT  X  13.457  3=  -f      1     4  .98 
A!Tr=-I~~15     8^52 
T  =  IP  13"*  12'.60 
T'=IQ  58      4^08 

This  repetition  will  be  rendered  unnecessary  by  always  giving 
the  rate  in  a  unit  of  the  clock.  Thus,  suppose  that  on  June  3, 
at  4*  11™  12s.  35  by  the  clock,  we  have  found  the  correction 
+  2m  10".14  ;  and  on  June  4,  at  14A  17™  49S.82,  we  have  found 
the  correction  +  2m  19*.89  ;  the  rate  in  one  hour  of  the  clock  will  be 


dT=        L       =  +  0-.2858 
34.1104 


*  For  practical  details  respecting  the  care  of  clocks  and  other 
time-keepers,  the  methods  of  comparing  their  indications,  &c., 
see  Vol.  II. ;  see  also  Chapter  VII.,.  u  Longitude  by  Chronometer." 
I  shall  here  confine  myself  to  the  methods  of  determining  their 
correction  by  astronomical  observation. 

Those  methods,  however,  which  involve  details  depending 
upon  the  peculiar  nature  of  the  instrument  with  which  the  ob- 
servation is  made,  will  be  treated  very  briefly  in  this  chapter, 
and  their  full  discussion  will  be  reserved  for  Vol.  II. 


196  TIME. 


FIRST   METHOD. — BY   TRANSITS. 

138.  At  the  instant  of  a  star's  passage  over  the  meridian,  note 
the  time  T\>y  the  clock.  The  star's  hour  angle  at  that  instant 
is  =  0A,  whence  the  local  sidereal  time  T'  is  (Art.  55) 

T'  =  a,  =  the  star's  right  ascension. 

If  the  clock  is  regulated  to  the  local  sidereal  time,  we  have, 
therefore, 

&T=a—  T 

But  if  the  clock  is  regulated  to  the  local  mean  time,  we  first  con- 
vert the  sidereal  time  a  into  the  corresponding  mean  time  T' 
(Art.  52),  and  then  we  have 

*T=  T'—T 

This,  then,  is  in  theory  the  simplest  and  most  direct  method 
possible.  It  is  also  practically  the  most  precise  when  properly 
carried  out  with  the  transit  instrument.  But,  as  the  transit  in- 
strument is  seldom,  if  ever,  precisely  adjusted  in  the  meridian, 
the  clock  time  T  of  the  true  meridian  transit  of  a  star  is  itself 
deduced  from  the  observed  time  of  the  transit  over  the  instru- 
ment by  applying  proper  corrections,  the  theory  of  which  will 
be  fully  discussed  in  Vol.  II. 

It  will  there  be  seen,  also,  that  the  time  may  be  found  from 
transits  over  any  vertical  circle. 


SECOND    METHOD. — BY   EQUAL    ALTITUDES. 

» 

139.  (A.)  Equal  altitudes  of  a  fixed  star. — The  time  of  the  meri- 
dian transit  of  a  fixed  star  is  the  mean  between  the  two  times 
when  it  is  at  the  same  altitude  east  and  west  of  the  meridian ;  so 
that  the  observation  of  these  two  times  is  a  convenient  substi- 
tute for  that  of  the  meridian  passage  when  a  transit  instrument 
is  not  available.  The  observation  is  most  frequently  made  with 
the  sextant  and  artificial  horizon ;  but  any  instrument  adapted  to 
the  measurement  of  altitudes  may  be  employed.  It  is,  however, 
not  required  that  the  instrument  should  indicate  the  true  alti- 
tude ;  it  is  sufficient  if  the  altitude  is  the  same  at  both  observa- 


BY    EQUAL   ALTITUDES.  197 

tions.  If  we  use  the  same  instrument,  and  take  care  not  to 
change  any  of  its  adjustments  between  the  two  observations,  we 
may  generally  assume  that  the  same  readings  of  its  graduated 
arc  represent  the  same  altitude.  Small  inequalities,  however, 
may  still  exist,  which  will  be  considered  hereafter.* 

The  clock  correction  will  be  found  directly  by  subtracting 
the  mean  of  the  two  clock  times  of  observation  from  the  com- 
puted time  of  the  star's  transit. 

EXAMPLE  1. — March  19,  1856 ;  an  altitude  of  Arcturus  east 
of  the  meridian  was  noted  at  II71  4W  51". 5  by  a  sidereal  clock, 
and  the  same  altitude  west  of  the  meridian  at  17/l  2lm  30*.0;  find 
the  clock  correction. 

East  11*    4-51-.5 

West  17   21    30.0 


Merid.  transit  by  clock  ==  T  =  14  13  10  .75 
March  19,  Arcturus  B.  A  =  a  =14  9  7.11 
Clock  correction  ==  A  T  =  -  4  3  .64 

This  is  the  clock  correction  at  the  sidereal  time  14*  9m  7s.ll  or 
at  the  clock  time  14*  13W  10-.75. 

EXAMPLE  2.— March  15,  1856,  at  the  Cape  of  Good  Hope, 
Latitude  33°  56'  S.,  Longitude  1*  13W  56s  E. ;  equal  altitudes  of 
Spica  are  observed  with  the  sextant  as  below,  the  times  being 
noted  by  a  chronometer  regulated  to  mean  Greenwich  time. 
The  artificial  horizon  being  employed,  the  altitudes  recorded  are 
double  altitudes. 

East.  2  Alt.  Spica.                    West. 

10*  20"*    0-.5  104°    0'  2h  40™  38s. 

«    20    28.  «      10  «  40    10.5 

"     20    55.  "      20  «  39    42. 


Means  10   20    27  .83  2  40    10  .17 

10  20    27.83 
Merid.  Transit,  by  Chronom.  =  T  =  12  30    19  .00 

The  chronometer  being  regulated  to  Greenwich  time,  we 
must  compute  the  Greenwich  mean  time  of  the  star's  transit  at 
the  Cape  (Art.  52).  We  have 

*  For  the  method  of  observing  equal  altitudes  with  the  sextant,  see  Vol.  II., 
"Sextant." 


198  TIME. 

Local  sidereal  time  of  transit  =  a  =  13*  17m  37*. 92 

Longitude                                            =  —    1  13    56 . 

Greenwich  sidereal  time                    =  12  3    41  .92 

March  15,  sid.  time  of  mean  noon  =  23  33      5  .37 

Sid.  interval  from  mean  noon           =  12  oO    36  .55 

Reduction  to  mean  time                    =  —  22 .97 
Mean  Gr.  time  of  star's  -» 

local  transit                }     =   T>  =  12  28    33.58 

Chronometer  time  of  do.       =    T  =       12   30    19  .00 
Chronometer  correction        =  A  T—  —          1     45  .42 

140.  (B).  Equal  altitudes  of  the  sun  before  and  after  noon. — If  the 
declination  of  the  sun  were  the  same  at  both  observations,  the 
hour  angles  reckoned  from  the  meridian  east  and  west  would  be 
equal  when  the  altitudes  were  equal,  and  the  mean  of  the  two 
clock  times  of  observation  would  be  the  time  by  the  clock  at 
the  instant  of  apparent, noon,  and  we  should  find  the  clock  cor- 
rection as  in  the  case  of  a  fixed  star.  To  find  the  correction 
for  the  change  of  declination,  let 

y  =  the  latitude  of  the  place  of  observation, 
8  =.  the  sun's  declination  at  apparent  (local)  noon, 
A<S  =  the  increase  of  declination  from  the  meridian  to  the  west 

observation,  or  the  decrease  to  the  east  observation, 
h  =  the  sun's  true  altitude  at  each  observation, 
T(>  =  the  mean  of  the  clock  times  A.M.  and  P.M., 
A  TQ  =  the  correction  of  this  mean  to  reduce  to  the  clock  time 

of  apparent  noon, 
t  =  half  the  elapsed  time  between  the  observations. 

Then  we  have 

t  -f-  A  T0  —  the  hour  angle  at  the  A.  M.  observation  reckoned 

towards  the  east, . 

t  —  &T0  =  the  hour  angle  at  the  P.M.  observation, 
<5  —  A#    =  the  declination  at  the  A.M.         " 
8  -f  A<?    =  "  "     P.M          « 

and,  by  the  first  equation  of  (14)  applied  to  each  observation, 

sin  h  =  sin  <p  sin  (d  —  A<5)  -\-  cos  <p  cos  (d  —  AO)  cos  (t  -f-  A  ^o) 
sin  h  =  sin  <p  sin  (<5  -f  A#)  -f  cos  ^  cos  (d  -\-  A'3)  cos  (t  —  A  TQ) 


BY    EQUAL  ALTITUDES.  199 

If  we  substitute 

sin  (d  ±:  Ad)  =  sin  d  cos  Ad  ±  cos  d  sin  Ad 
cos  (d  ±  Ad)  =  cos  d  cos  Ad  =F  sin  d  sin  Ad 
cos  (t  it  A  Ty)  =  cos  t  cos  A  !T0  qp  sin  £  sin  A  7^ 

and  then  subtract  the  first  equation  from  the  second,  we  shall 
find 

0  =  2  sin  tp  cos  d  sin  Ad  —  2  cos  y  sin  d  sin  Ad  cos  £  cos  A  T0 
-f-  2  cos  <p  cos  d  sin  t  cos  Ad  sin  A  T0 

whence,  by  transposing  and  dividing  by  the  coefficient  of  sin  A7"0, 

tan  Ad  .  tan  y        tan  Ad  .  tan  d 

sin  A  Tft  = cos  A  TQ 

sin  t  tan  t 

This  is  a  rigorous  expression  of  the  required  correction  ±TW  but 
the  change  of  declination  is  so  small  that  we  may  put  Ad  for  its 
tangent,  &TQ  for  its  sine,  and  unity  for  cos  A  Tw  without  any 
appreciable  error ;  and,  since  Ad  is  expressed  in  seconds  of  arc, 
we  shall  obtain  &T0  in  seconds  of  time  by  dividing  the  second 
member  by  15.  We  thus  find  the  formula* 

*r°  =  -^nT  +  irS7  (262) 

The  Ephemeris  gives  the  hourly  change  of  d.  If  we  take  it  for 
the  Greenwich  instant  corresponding  to  the  local  noon,  and  call 
it  A  'd,  and  if  t  is  reduced  to  hours,  we  have 

and  our  formula  becomes 

A  T  =  _  A/(*-nan?  1  A''K*tan<*        [Equation!       (263 
15  sin  t  15  tan  t  Lfor  noon.J 

To  facilitate  the  computation  in  practice,  we  put 

A     _  7>    

15  sin/  15  tan  t 

a=A.A'd.tim<p          b  =  B .  A'd .  tan  d  (264) 

then  we  have 


*  As  first  given  by  GAUSS,  Monatliche  Correspondenz,  Vol.  23. 


200  TIME. 

The  correction  &TQ  is  called  the  equation  of  equal  altitudes.  The 
computation  according  to  the  above  form  is  rendered  extremely 
simple  by  the  aid  of  our  Table  IV.,  which  gives  the  values  of 
log  A  and  log  B  with  the  argument  "elapsed  time"  (=  2/). 
Then  a  and  b  are  computed  as  above,  the  algebraic  signs  of  the 
several  factors  being  duly  observed.  When  the  sun  is  moving 
towards  the  north,  give  &'d  the  positice  sign ;  and  also  when 
<p  and  d  are  north,  give  them  the  positive  sign ;  in  the  opposite 
cases  they  take  the  negative  sign.  The  signs  of  A  and  B  are 
given  in  'the  table ;  A  being  negative  only  when  /  <  12*  and  B 
positive  when  t  <  6h  or  >  18*. 

When  we  have  applied  A  T0  to  the  mean  of  the  clock  times  (or 
the  "  middle  time"),  we  have  the  time 

T=  TO  +  AT; 

as  shown  by  the  clock  at  the  instant  of  the  sun's  meridian  transit. 
Then,  computing  the  time  T',  whether  mean  or  sidereal,  which 
the  clock  is  required  to  show  at  that  instant,  wre  have  the  clock 
correction,  as  before, 

Ar=  T  —  T 

EXAMPLE. — March  5,  1856,  at  the  U.  S.  Naval  Academy,  Lat. 
38°  59'  K,  Long.  5*  5"1  57*.5  W.,  the  sun  was  observed  at  the 
same  altitude,  A.M.  and  P.M.,  by  a  chronometer  regulated  to 
mean  Greenwich  time ;  the  mean  of  the  A.M.  times  was  1*  8'"  26*. 6, 
ancj  of  the  P.M.  times  87t  45nt  41*.7 ;  find  the  chronometer  cor- 
rection at  noon. 

We  have  first  A.M.  Chro.  Time  =  1*    8™  26'.6 

P.M.  «  «  =  8  45  41  .7 
Elapsed  time  2t  =  7  37  15.1 
Middle  time  TQ  =  4  57  4~15 

From  the  Ephemeris  we  find  for  the  local  apparent  noon  of- 
March  5,  1856, 

<5  =  _  50  46'  22".5        Equation  of  time  =  +  llm  35M1 
A'<$  ==  +  58'MO 

For  the  utmost  precision,  we  reduce  A'£  to  the  instant  of  local 


BY    EQUAL    ALTITUDES.  201 

noon.  With  these  quantities  and  <p  =  38°  59',  we  proceed  as 
follows: 

Arg.  7*  37™  Table  IV.  log  A        w9.4804  log  B         9.2151 

log  A'<5        1.7642  log  A'<*       1.7642 

log  tan  <p   9.9081  log  tan  3  n9.0047 

log  a        nl.1527  log  b        n9  9840 

a  =  —  14-.21  b  =  —  0-.96 

Middle  Chro.  time  T0  =  4*  57"  4U5 

A  T0  =  a  -f-  b  =        —  15.17 

Chro.  Time  of  app.  noon  T  =  4  56    48.98 

This  quantity  is  to  be  compared  with  the  Greenwich  time  of  the 
local  apparent  noon,  since  the  chronometer  is  regulated  to 
Greenwich  time.  We  have 

Mean  local  time  of  app.  noon          =  0*  llw  35'.11 

Longitude  =  5     5    57  .50 

Mean  Greenwich  time         «  =  T'  =  5   17    32  .61 

A  T  =  T'  —  T  =  -f  20m  43^63 

If  the  correction  of  the  chronometer  to  mean  local  time  is 
required,  we  have  only  to  omit  the  application  of  the  longitude. 
Thus,  we  should  have 

Chro.  time  of  app.  noon     —  4*  56m  48*. 98 
Equation  of  time  —  —  11    35  .11 

Chro.  time  of  mean  noon  =  4    45    13  .87 

and  since  at  mean  noon  a  chronometer  regulated  to  the  local 
time  should  give  Oh  Om  0",  it  is  here  fast,  and  its  correction  to 
local  time  is  —  4*  45™  13'.87. 


141.  (C.)  Equal  altitudes  of  the  sun  in  the  afternoon  of  one  day  and 
the  morning  of  the  next  following  day  ;  i.e.  before  and  after  midnight. — 
It  is  evident  that  when  equal  zenith  distances  are  observed  in 
the  latitude  -f-  ^,  their  supplement  to  180°  may  be  considered  as 
equal  zenith  distances  observed  at  the  antipode  in  latitude  —  <p 
on  the  same  meridian.  Hence  the  formula  (263)  will  give  the 
equation  for  noon  at  the  antipode  by  substituting  —  <p  for  -f-  ^, 
that  is,  by  changing  the  sign  of  the  first  term ;  but  this  noon  at 


202  TIME. 

the  antipode  is  the  same  absolute  instant  as  the  midnight  of  the 
observer,  and  hence 


T  _  A'a  .  t  tan  <?        b'd.ttund       ("Equation  for"] 
15  sin  t  15  tan  t        L    midnight.   J 

and  this  is  computed  with  the  aid  of  the  logarithms  of  A  and  B 
in  Table  IY.  precisely  as  in  (264),  only  changing  the  sign  of  A. 
The  sign  for  this  case  is  given  in  the  table.* 

142.  To  find  the  correction  for  small  inequalities  in  the  altitudes.  — 
If  from  a  change  in  the  condition  of  the  atmosphere  the  re- 
fraction is  different  at  the  two  observations,  equal  apparent  alti- 
tudes will  not  give  equal  true  altitudes.  To  find  the  change  &t 
in  the  hour  angle  t  produced  by  a  change  A/I  in  the  altitude  h, 
we  have  only  to  differentiate  the  equation 

sin  h  =  sin  </>  sin  8  -f-  cos  <p  cos  d  cos  t 
regarding  <p  and  d  as  constant  ;  whence 

cos  h  .  A/I  =  —  cos  <p  cos  d  sin  t  . 


where  AA  is  in  seconds  of  arc  and  A£  in  seconds  of  time. 

If  the  altitude  at  the  west  observation  is  the  greater  by  A/i,  the 
hour  angle  is  increased  by  A£,  and  the  middle  time  is  increased  by 
\  A/.     The  correction  for  the  difference  of  altitudes  is  therefore 
-  \  A£,  and,  denoting  it  by  A'  Tw  we  have,  by  the  above  equation, 

(266) 


. 
30  cos  <p  cos  d  sin  t 

This  correction  is  to  be  added  algebraically  to  the  middle  clock 
time  in  any  of  the  cases  (A),  (B),  (C)  of  the  preceding  articles. 

EXAMPLE.  —  Suppose  that  in  Example  2,  Art.  139,  there  had 
been  observed  at  the  east  observation  Barom.  30.30  inches, 
Therm.  35°  F.,  but  at  the  west  observation  Barom.  29.55  inches, 
Therm.  52°  F.  We  have  for  the  altitude  52°  5'  or  zenith  dis- 
tance 37°  55',  by  Table  L,  the  mean  refraction  45".4.  By  Table 

*  For  an  example  and  some  practical  remarks,  see  my  "Improved  method  of 
finding  the  error  and  rate  of  a  chronometer  by  equal  altitudes,"  Appendix  to  the 
American  Ephemeris  for  1856  and  1857. 


BY    EQUAL   ALTITUDES.  203 

XIY.A  and  XIY.B,  the  corrections  for  the  barometer  and  ther- 
mometer are  as  follows,  taking  for  greater  accuracy  one-eighth 
of  the  corrections  for  6' : 

East  Obs.  West  Obs. 

Barom.  30.30  +  0".5  Barom.  29.55  —  0".6 

Therm.  35°.  -}- 1  A  Therm.  52°.  -0.1  « 

•+  1  .9  -  0  .7 

The  difference  of  these  numbers  gives  A/I  =  +  2". 6  as  the  excess 
of  the  true  altitude  at  the  west  observation.  Hence,  by  the 

formula  (266), 

A/I  =  +    2".6  log  A/I  0.415 

h  =       52°    5'  log  cos  h  9.789 

y  =  —  33    56  log  sec  y>  0.081 

d  =  —  10    25  log  sec  3  0.007 

t  =  J  elapsed  time  =  2*  9m    51'.  log  cosee  t  0.270    . 

log  a'0  ^.523 

AT0  =  +  OM2  logA'T;  9.085 

When,  however,  several  altitudes  have  been  observed,  as  in 
this  example,  we  may  obtain  this  correction  from  the  observa- 
tions themselves ;  for  we  see  that  the  double  altitude  of  Spica 
changed  20'  =  1200"  in  about  55*,  and  hence  we  have  the 

proportion 

1200"  :2".6  =  55';  A'  Tn 


0 


which  gives  A'  T0  =  -f  0'.12  as  before.  By  taking  the  change  in 
the  double  altitude,  the  fourth  term  is  the  value  of  J  A/,  or  A'  Tv 

If  this  correction  be  applied,  we  find  the  corrected  time  of 
transit  =  12A  30W  19".  12,  and  consequently  the  chronometer  cor- 
rection A  T=  —  lm  45'.54. 

The  altitudes  may  differ  from  other  causes  besides  a  change  in 
the  refraction;  for  instance,  the  second  observation  may  be  in- 
terrupted by  passing  clouds,  so  that  the  precisely  corresponding 
altitude  cannot  be  taken ;  but,  rather  than  lose  the  whole  ob- 
servation, if  we  can  observe  an  altitude  differing  but  little  from 
the  first,  we  may  use  it  as  an  equal  altitude,  and  compute  the 
correction  for  the  difference  by  the  formula  (266). 

143.  Effect  of  errors  in  the  latitude,  decimation,  and  altitude  upon 
the  time  found  by  equal  altitudes. — The  time  found  by  equal  altitudes 
of  a  fixed  star  is  wholly  independent  of  errors  in  the  latitude 


204  TIME. 

and  declination,  since  these  quantities  do  not  enter  into  the  com- 
putation. In  observations  of  the  sun,  an  error  in  the  latitude 
affects  the  term 

a  =  A  A'(S  tun  <p 

by  differentiating  which  we  find  that  an  error  d<p  produces  in  a 
the  error  da  =  A  &'S  .  sec2  <f> .  dy,  or,  putting  sin  dtp  for  dp, 

da  =  A  A'O  sec2  <f  sin  dtp 

In  the  same  manner,  we  find  that,  an  error  dd  in  the  declination 
produces  in  6  the  error 

db  =  £&'3  sec2  <5  sin  dd 

In  the  example  of  Art.  140,  suppose  the  latitude  and  declina- 
tion were  each  in  error  V.  We  have 

log  A  A'<5  nl.2446  log  B*'d   0.9793 

log  sec2  <f>    0.2188  log  sec2  d  0.0044 

log  sin  1'     6.4637  log  sin  1'   6.4637 

log  da       w7.9271  logdb        7.4474 
da  =  —  O'.OOS  db  =  -f  0«.003 

If  dtp  and  dd  had  opposite  signs,  the  whole  error  in  this  case  would 
be  O'.OOS  +  0".003  =  O'.Oll.  As  the  observer  can  always  easily 
obtain  his  latitude  within  V  and  the  declination  (even  when  the 
longitude  is  somewhat  uncertain)  within  a  few  seconds,  we  may 
regard  the  method  as  practically  free  from  the  effects  of  any 
errors  in  these  quantities.  The  accuracy  of  the  result  will  there- 
fore depend  wholly  upon  the  accuracy  of  the  observations. 

The  accuracy  of  the  observations  depends  in  a  measure  upon 
the  constancy  of  the  instrument,  but  chiefly  upon  the  skill  of  the 
observer.  Each  observer  may  determine  the  probable  error  of 
his  observations  by  discussing  them  by  the  method  of  least 
squares.  An  example  of  such  a  discussion  will  be  given  in  the 
following  article. 

The  effect  of  an  error  in  the  altitude  is  given  by  (266).  Since 
we  have,  A  being  the  azimuth  of  the  object, 

cos  d  sin  t 

sin  A  =  - 

cos  h 


BY    EQUAL   ALTITUDES.  205 

the  formula  may  also  be  written 


30  cos  <p  sin  A 

which  will  be  least  when  the  denominator  is  greatest,  i.e.  when 
A  —  90°  or  270°,  or  when  the  object  is  near  the  prime  vertical. 
From  this  we  deduce  the  practical  precept  to  take  the  observations 
when  the  object  is  nearly  east  or  west.  This  rule,  however,  must  not 
be  carried  so  far  as  to  include  observations  at  very  low  altitudes, 
where  anomalies  in  the  refraction  may  produce  unknown  dif- 
ferences in  the  altitudes.  If  the  star's  declination  is  very  nearly 
equal  to  the  latitude,  it  will  be  in  the  prime  vertical  only  when 
quite  near  to  the  meridian,  and  then  both  observations  may  be 
obtained  within  a  brief  interval  of  time;  and  this  circumstance 
is  favorable  to  accuracy,  inasmuch  as  the  instrument  will  be  less 
liable  to  changes  in  this  short  time. 


144.  Probable  error  of  observation.  —  The  error  of  observation  is 
composed  of  two  errors,  one  arising  from  imperfect  setting  of 
the  index  of  the  sextant,  the  other  from  imperfect  noting  of  the 
time;  but  these  are  inseparable,  and  can  only  be  discussed  as  a 
single  error  in  the  observed  time.  The  individual  observations 
are  also  affected  by  any  irregularity  of  graduation  of  the  sextant, 
but  this  error  does  not  affect  the  mean  of  a  pair  of  observations 
on  opposite  sides  of  the  meridian;  and  therefore  the  error  of 
observation  proper  will  be  shown  by  comparing  the  mean  of 
the  several  pairs  with  the  mean  of  these  means.  If,  then,  the 
mean  of  a  pair  of  observed  times  be  called  a^  the  mean  of  all 
these  means  aw  the  probable  error  of  a  single  pair,  supposing  all 
to  be  of  the  same  weight,  is* 


r- 


in  which  n  =  the  number  of  pairs,  and  q  =  0.6745  is  the  factor 
to  reduce  mean  to  probable  errors.     The  probable  error  of  the 


final  mean  «0  is 


*  See  Appendix,  Least  Squares. 


206 


TIME. 


EXAMPLE.  —  At  the  U.  S.  Naval  Academy,  June  18,  1849,  the 
following  series  of  equal  altitudes  of  the  sun  was  observed. 


Chro.  A.M. 

Chro.  P.M. 

a 

a 

,_ 

«o 

0A  43™  53*. 

9*  44™  3'.  5   5*  13™  58*.  25 

-f 

CM  2 

44 

19. 

43 

38 

. 

58 

.50 

+ 

0. 

37 

44 

45. 

43 

11 

.5 

58 

.25 

+ 

0. 

12 

45 

11. 

42 

46 

.3 

58 

.65 

+ 

0.52 

45 

37. 

42 

19 

.7 

58 

.35 

+ 

0. 

22 

46 

1  .7 

41 

53 

.5 

57 

.60 

— 

0. 

53 

46 

28.5 

41 

27 

. 

57 

.75 

— 

0. 

38 

46 

55. 

41 

0 

.5 

57 

.75 

— 

0. 

38 

47 

19.7 

40 

36 

.5 

58 

.10 

— 

0. 

03 

a0  =  5  13 

58 

.13 

2 

(a 

—  ao)2 

n 

=  9 

f(a 

-*o)2 

(a  -  flo> 
0.0144 
.1369 
.0144 
.2704 
.0484 
.2809 
.1444 
.1444 
.0009 

=  1.0551 


rft  =  -  —     =      0'.082 
</n 

A  similar  discussion  of  a  number  of  sets  of  equal  altitudes  of 
the  sun  taken  by  the  same  observer  gave  O.23  as  the  probable 
error  of  a  single  pair  for  that  observer,  and  consequently  the 
probable  error  of  the  result  of  six  observations  on  each  side  of 
the  meridian  would  be  only  0*.23  -f-  v/  6  =  0\094.  This,  how- 
ever, expresses  only  the  accidental  error  of  observation,  and  does 
not  include  the  effect  of  'changes  in  the  state  of  the  sextant  be- 
tween the  morning  and  afternoon  observations.  Such  changes 
are  not  unfrequently  produced  by  the  changes  of  temperature  to 
which  it  is  exposed  in  observations  of  the  sun  ;  it  is  important, 
therefore,  to  guard  the  instrument  from  the  sun's  rays  as  much 
as  possible,  and  to  expose,  it  only  during  the  few  minutes 
required  for  each  observation.  The  determination  of  the  time 
by  stars  is  mostly  free  from  difficulties  of  this  kind,  but  the 
observation  is  not  otherwise  so  accurate  as  that  of  the  sun,  ex- 
cept in  the  hands  of  very  skilful  observers. 


THIRD    METHOD. — BY   A    SINGLE    ALTITUDE,    OR    ZENITH    DISTANCE. 

145..  Let  the  altitude  of  any  celestial  bod}r  be  observed  with 
the  sextant  or  any  altitude  instrument,  and  the  time  noted  by 
the  clock.  For  greater  precision,  observe  several  altitudes  in 
quick  succession,  noting  the  time  of  each,  and  take  the  mean  of 
the  altitudes  as  corresponding  to  the  mean  of  the  times.  But 


BY    A    SINGLE    ALTITUDE.  207 

in  taking  the  mean  of  several  observations  in  this  way,  it  must 
not  be  forgotten  that  we  assume  that  the  altitude  varies  in  pro- 
portion to  the  time,  which  is  theoretically  true  only  in  the 
exceptional  case  where  the  observer  is  on  the  equator  and  the 
star's  declination  is  zero.  It  is,  however,  practically  true  for  an 
interval  of  a  few  minutes  when  the  star  is  not  too  near  the 
meridian.  The  observations  themselves  will  generally  show  the 
limit  beyond  which  it  will  not  be  safe  to  apply  this  rule.  When 
the  observations  have  been  extended  beyond  this  limit,  a  cor- 
rection for  the  unequal  change  in  altitude  (i.e.  for  second  differ- 
ences) can  be  applied,  which  will  be  treated  of  below. 

With  the  altitude  and  azimuth  instrument  we  generally  ob- 
tain zenith  distances  directly.  In  all  cases,  however,  we  may 
suppose  the  observation  to  give  the  zenith  distance.  Having 
then  corrected  the  observation  for  instrumental  errors,  for  re- 
fraction, £c.,  Arts.  135,  136,  letjM)e  the  resulting  true  or  geo- 
centric zenith  distance.  Let  <p  be  the  latitude  of  the  place  of 
observation,  3  the  star's  decimation,  t  the  star's  hour  angle. 
The  three  sides  of  the  spherical  triangle  formed  by  the  zenith, 
the  pole,  and  the  star  may  be  denoted  by  a  =  90°  —  ^,  b  =  £,  c  = 
90°  —  o,  and  the  angle  at  the  pole  by  13  =  t,  and  hence,  Art.  22, 
we  deduce 

siu  tt=  Ji  S'"  *  [C  +  (<f  -  S^  8i"  *  E  -(»  -  *fl  )        (207) 

\  \  COS  </>  COS  S  / 

which  gives  t  by  a  very  simple  logarithmic  computation.  From 
t  we  deduce,  by  Art.  55,  the  local  time,  which  compared  with 
the  observed  clock  time  gives  the  clock  correction  required. 

It  is  to  be  observed  that  the  double  sign  belonging  to  the 
radical  in  (267)  gives  two  values  of  sin  J  /,  the  positive  corre- 
sponding to  a  west  and  the  negative  to  an  east  hour  angle;  since 
any  given  zenith  distance  may  be  observed  on  either  side  of  the 
meridian.  To  distinguish  the  true  solution,  the  observer  must 
of  course  note  on  which  side  of  the  meridian  he  has  observed. 

If  the  object  observed  is  the  sun,  the  moon,  or  a  planet,  its 
declination  is  to  be  taken  from  the  Ephemeris,  for  the  time  of 
the  observation  (referred  to  the  meridian  of  the  Ephemeris);  but, 
as  this  time  is  itself  to  be  found  from  the  observation,  we  must 
at  first  assume  an  approximate  value  of  it,  with  which  an  approxi- 
mate declination  is  found.  With  this  declination  a  first  compu- 


208 


TIME. 


tation  by  the  formula  gives  an  approximate  value  of  £,  and  hence 
a  more  accurate  value  of  the  time,  and  a  new  value  of  the  decli- 
nation, with  which  a  second  computation  by  the  formula  gives  a 
still  more  accurate  value  of  t.  Thus  it  appears  that  the  solution 
of  our  problem  is  really  indirect,  and  theoretically  involves  an 
infinite  series  of  successive  approximations;  in  practice,  how- 
ever, the  observer  generally  possesses  a  sufficiently  precise  value 
of  his  clock  correction  for  the  purpose  of  taking  out  the  declina- 
tion of  the  sun  or  planets.  The  moon  is  never  employed  for 
determining  the  local  time  except  at  sea,  and  when  no  other 
object  is  available.* 

EXAMPLE.  —  At  the  U.  S.  Naval  Academy,  in  Latitude  <p  =  38° 
58'  53"  K,  Longitude  5*  5W  57*.5  W.,  December  9,  1851,  the  fol- 
lowing double  altitudes  of  the  sun  west  of  the  meridian  were 
observed  with  a  sextant  and  artificial  horizon,  the  times  being 
noted  by  a  Greenwich  mean  time  chronometer: 


Chronometer. 

7*  35-  14-.5  33°  30'  Barom.  30.28  inches. 

35  55.  «  20  Att.  Therm.  55°  F. 

36  35.5  "  10  Ext.  Therm.  50°  F. 

37  15  .5  "  0  Index  correction  of  the 
37    55.  32  50  sextant  =  —  1'  10" 

Means  7~~36    35.1  33  10 

The*  approximate  correction  of  the  chronometer  was  assumed  to 
be  -f-  9m  40*.     Find  its  true  correction. 

With  the  assumed  chronometer  correction  we  obtain  the  ap- 
proximate Greenwich  time  =  lh  46™  15s,  with  which  we  take 
from  the  Ephemeris 

d  =  —  22°  50'  27"  Sun's  scmidiametcr  S  =  16'  17" 
Eq.  of  time   =  —    7m  25'.80       "       hor.  parallax    -x  =          8".7 

We  have  then 


*  But  the  moon's  altitude  and  the  hour  angle  deduced  from  it  may  be  used  in 
finding  the  observer's  longitude,  as  will  be  shown  in  the  Chapter  on  Longitude. 

f  The  symbol  Q  is  used  for  "observed  altitude  of  the  sun's  lower  limb,"  and  2  Q 
for  the  double  altitude  from  the  artificial  horizon.  In  a  similar  manner  we  use 

O,  X  T- 


BY    A    SINGLE    ALTITUDE. 


209 


Observed  20=  33°  10'    0" 

Index  corr.       =  —  1  10 

33  8  50 

App.  altitude    =  16  34  25 

z  =  73  25  35 

(Table  II.)  r  =  +  3  15 

TT  smz  =  p  =  —  8 

S=-  16  17 

C  =  73  12  25 

The  computation  by  (267)  is  then  as  follows : 


Library 


Calif 


S  = 


38°  58'  53" 
—  22    50   27 


61 
73 


49 
12 


20 
25 


£  sum  = 
*  diff.  = 


67    30   52  .5 
5   41   32  .5 


log  sec  <p  0.109383 
log  sec  S  0.035464 
log  sin  1  sum  9.965661 
log  sin  J  diff.  8.996455 
19.106963 


log  sin  %t         9.553482 

J  *  =  20°  57'  25".6 
Apparent  time  =  t  =  2*  47™  39'.  4 
Eq.  of  time  -   7    25  .8 

Local  mean  time      =  2  40    13  .6 
Longitude  =5     5    57 .5 


True  Gr.  Time  =  T'  =  1  46    11.1 
T=7   36    35.1 


=  +   9    36  .0 

agreeing  so  nearly  with  the  assumed  correction  that  a  repetition 
of  the  computation  is  unnecessary. 

146.  If  it  is  preferred  to  use  the  altitude  instead  of  the  zenith 
distance,  put  the  true  altitude  h  =  90°  —  £,  and  the  polar  distance 
of  the  star  P  =  90°  —  3,  then  we  have,  in  (267), 

sin  J  [C  —  (?  — *)]=8in  J(90°  —  h  —  ^-f  90°  —  P)  =  cos  J(ft+p-f  P) 
If  then  we  put 


the  formula  becomes 

sin  }  t  = 

VOL.  I.— 14 


cos  s  sin  (s  — 
cos      sin  P 


(268) 


210  TIME. 

In  this  form  we  may  always  take  P  =  the  distance  from  the  ele- 
vated pole,  and  regard  the  latitude  as  always  positive,  and  then 
no  attention  to  the  algebraic  signs  of  the  quantities  in  the  second 
member  is  required.  Thus,  in  the  preceding  example,  we  should 
proceed  as  follows : 

App.  alt.  ==    16°  34'  25" 
r— p=    —     3     7 
8=          16  17 
A  =    16    47  35 

<f>  =    38    58  53      log  sec      0.109383 

P  =  112    50   27      log  cosec  0.035464 

25  =  168    36   55 

s  =    84    18  27  .5 log  cos      8.996455 

$  —  h=    67    30   52  .5 log  sin      9.965661 

19.106963 

and  the  computation  is  finished  as  in  the  preceding  article. 

147.  If  we  aim  at  the  greatest  .degree  of  precision  which  the 
logarithmic  tables  can  afford,  we  should  find  the  angle  J£  by  its 
tangent,  since  the  logarithms  of  the  tangent  always  vary  more 
rapidly  than  those  of  the  other  functions.  For  this  purpose  we 
deduce 

^=i(C  +  f-f 

(269) 


cos  s  cos  (s  — 
or,  if  the  altitude  is  used, 

s  =  J  (A  -f-  <p  -f  P) 

cos  s  sin  (s  —  A) 


'S  —  y)  COS  (S-  —  P) 


(270) 


148.  If  a  number  of  observations  of  the  same  star  at  the  same 
place  are  to  be  individually  computed,  it  will  be  most  readily 
done  by  the  fundamental  equation 

cos  C  —  sin  <p  sin  3 
cos  l  =  - 

cos  <f>  cos  8 


BY    A    SINGLE    ALTITUDE.  211 

for  the  logarithms  of  sin  y  sin  o  and  cos  <p  cos  d  will  be  constant, 
and  for  each  observation  we  shall  only  have  to  take  from  the 
trigonometric  table  the  log.  of  cos  £  ;  the  logarithm  of  the  nume- 
rator will  then  be  found  by  the  aid  of  ZECH'S  Addition  or  Sub- 
traction Table,  which  is  included  in  HULSSE'S  edition  of  VEGA'S 
Tables.  The  addition  or  the  subtraction  table  will  be  used  ac- 
cording as  sin  <p  sin  3  is  positive  or  negative. 

149.  Effect  of  errors  in  the  data  upon  the  time  computed  from  an 
altitude.  —  We  have  from  the  differential  equation  (51),  Art.  35, 
multiplying  dt  by  15  to  reduce  it  to  seconds  of  arc, 

sin  q  cos  d  (15  dt)  =  d*  —  cos  A  dtp  -\-  cos  q  d3 

where  </£,  dy>,  do,  may  denote  small  errors  of  £,  <p,  <5,  and  dt  the 
corresponding  error  of  t  ;  A  is  the  star's  azimuth,  q  the  parallactic 
angle,  or  angle  at  the  star. 

If  the  zenith  distance  alone  is  erroneous,  we  have,  by  putting 
d<p  =  0,  and  dd  —  0, 


15  dt  = 


sin  q  cos  d       cos  <p  sin  A 

from  which  it  follows  that  a  given  error  in  the  zenith  distance 
will  have  the  least  effect  upon  the  computed  time  when  the 
azimuth  is  90°  or  270° ;  that  is,  when  the  star  is  on  the  prime 
vertical;  for  we  then  have  sin  A  =  ±  1,  and  the  denominator 
of  this  expression  obtains  its  maximum  numerical  value.  Also, 
since  cos  <p  is  a  maximum  for  <p  =  0,  it  follows  that  observa- 
tions of  zenith  distances  for  determining  the  time  give  the 
most  accurate  results  when  the  place  is  on  the  equator.  On  the 
other  hand,  the  least  favorable  position  of  the  star  is  when  it  is 
on  the  meridian,  and  the  least  favorable  position  of  the  observer 
is  at  the  pole. 

By  putting  d£  —  0,  dd  =  0,  sin  q  cos  d  =  cos  <p  sin  A  we  have 

I5dt= 


cos  <p  tan  A 

by  which  we  see  that  an  error  in  the  latitude  also  produces  the 
least  effect  when  the  star  is  on  the  prime  vertical,  or  when  the 
observer  is  on  the  equator.  Indeed,  when  the  star  is  exactly  in 


212.  TIME. 

the  prime  vertical,  a  small  error  in  <p  has  no  appreciable  effect : 
since,  then,  tan  A  =  oo,  and  hence  when  the  latitude  is  uncertain, 
we  may  still  obtain  good  results  by  observing  only  stars  near  the 
prime  vertical. 

By  putting  d£  =  0,  dtp  =  0,  we  have 

d 


cos  d  tan  q 
which  shows  that  the  error  in  the  declination  of  a  given  star 

o 

produces  the  least  effect  when  the  star  is  on  the  prime  vertical  ;* 
and  of  different  stars  the  most  eligible  is  that  which  is  nearest 
to  the  equator. 

As  very  great  zenith  distances  (greater  than  80°)  are,  if  pos- 
sible, to  be  avoided  on  account  of  the  uncertainty  in  the  refraction, 
the  observer  will  often  be  obliged,  especially  in  high  latitudes, 
to  take  his  observations  at  some  distance  from  the  prime  vertical, 
in  which  case  small  errors  of  zenith  distance,  latitude,  or  declina- 
tion may  have  an  important  effect  upon  the  computed  clock  cor- 
rection. Nevertheless,  constant  errors  in  these  quantities  will 
have  no  sensible  effect  upon  the  rate  of  the  clock  deduced  from 
zenith  distances  of  the  same  star  on  different  days,  if  the  star  is 
observed  at  the  same  or  nearly  the  same  azimuth,  on  the  same 
side  of  the  meridian  ;  for  all  the  clock  corrections  will  be  in- 
creased or  diminished  by  the  same  quantities,  so  that  their 
differences,  and  consequently  the  rate,  will  be  the  same  as  if 
these  errors  did  not  exist.  The  errors  of  eccentricity  and 
graduation  of  the  instrument  are  among  the  constant  errors 
which  may  thus  be  eliminated. 

But  if  the  same  star  is  observed  both  east  and  west  of  the 
meridian,  and  at  the  same  distance  from  it,  sin  A  or  tan  J.,  and 
tan  7,  will  be  positive  at  one  observation  and  negative  at  the 
other,  and,  having  the  same  numerical  value,  constant  errors 
d^,  do,  and  d£  will  give  the  same  numerical  value  of  dt  with 
opposite  signs.  Hence,  while  one  of  the  deduced  clock  correc- 
tions will  be  too  great,  the  other  will  be  too  small,  and  their 
mean  will  be  the  true  correction  at  the  time  of  the  star's  transit 


*  From  the  equation  sin  q  =  — '. — -  sin  A,  it  follows  that  sin  q  is  a  maximum 

cos  6 

(for  constant  values  of  q>  and  J)  when  sin  A  =  1,  and  tan  q  is  a  maximum  in  tho 
same  case. 


CORRECTION    FOR    SECOND    DIFFERENCES.  213 

over  the  meridian.  Hence,  it  follows  again,  as  in  Art.  143,  that 
small  errors  in  the  latitude  and  declination  have  no  sensible 
effect  upon  the  time  computed  from  equal,  altitudes. 

150.    To  find  the  change  of  zenith  distance  of  a  star  in  a  given  in- 
teri'al  of  time,  having  regard  to  second  differences. 
The  formula 

d*  =  cos  <p  sin  A  dt 

is  strictly  true  only  when  d£  and  dt  are  infinitesimals.  But  the 
complete  expression  of  the  finite  difference  A£  in  terms  of  the 
finite  difference  &t  involves  the  square  and  higher  powers  of  &t. 
Let  £  he  expressed  as  a  function  of  t  of  the  form 

*==/• 

then,  to  find  any  zenith  distance  £  -f-  A£  corresponding  to  the 
hour  angle  t  -f  A<,  we  have,  by  TAYLOR'S  Theorem, 


or,  taking  only  second  differences, 


r  =  . 

dt  dt'1     2 


We  have  already  found 

Jn 

—  =  cos  <p  sm  A 
dt 

which  gives,  since  A  varies  with  £,  but  <p  is  constant, 

d2:  dA 

.  —  =  cos  <p  cos  A  •  — 

dt1  dt 

But  from  the  second  of  equations  (51)  we  have,  since  dd  and  dtp 
are  here  zero, 

dA       cos  q  cos  d       cos  q  sin  A 

dt  sin  C  sin  t 

whence 

d2C  _  .  cos  (p  sin  A  cos  A  cos  q 
dt'2  Bin  t 


214  TIME. 

and  the  expression  for  A£  becomes 

cos  <f>  sin  A  cos  A  cos  q  At2 

AC  =  cos  <f  sin  J.  .  A£  -j  --        -  *  -- 

sin  t  2 

Since  A£  and  A*  are  here  supposed  to  be  expressed  in  parts  of 
the  radius,  if  we  wish  to  express  them  in  seconds  of  arc  and  of 
time  respectively,  we  must  substitute  for  them  A£  sin  V  and 
15  Ai  sin  1",  and  the  formula  becomes 

cos  <p  sin  A  cos  A  cos  q   (15  Af)2  sin  1" 

A:=  costfsin^(15Af)  +  -  -_*.^__  (371) 

sin  t  2 

But  in  so  small  a  term  as  the  last  we  may  put 

(15  A*)2  sin  I"  _  2  sin2  \  &t  _ 
2  sin  I" 

the  value  of  which  is  given  in  our  Table  V.,  and  its  logarithm 
in  Table  VI.  ;  so  that  if  we  put  also 

cos  A  cos  q 

a  =  cos  <p  sin  .4,        k  =  —  —  -— 

sin  t 

we  shall  have 

akm  ,     272 


151.  A  number  of  zenith  distances  being  observed  at  yiien  clock 
times,  to  correct  the  mean  of  the  zenith  distances  or  of  the  clock  times 
for  second  differences.  —  The  iirst  term  of  the  above  value  of  A£ 
varies  in  proportion  to  A/,  but  the  second  term  varies  in  propor- 
tion to  A^2  ;  and  hence,  when  the  interval  is  sufficiently  great  to 
render  this  second  term  sensible,  equal  intervals  of  time  corre- 
spond to  unequal  differences  of  zenith  distance,  and  vice  versa  : 
in  other  words,  we  shall  have  second  differences  either  of  the 
zenith  distance  or  of  the  time.  Two  methods  of  correction 
present  themselves. 

1st.  Reduction  of  the  mean  of  the  zenith  distances  to  the  mean  of  the 
times.—  Let  Tv  T2,  7"3,  &c.  be  the  observed  clock  times  ;  £„  £2,  £3, 
&c.  the  corresponding  observed  zenith  distances;  Tihe  mean  of 
the  times;  £0  the  mean  of  the  zenith  distances;  £  the  zenith 
distance  corresponding  to  T.  The  change  d  —  £  corresponds  to 
the  interval  71,  —  T,  £2  —  C  to  T2  —  7}  &c.  ;  so  that  if  we  put 


CORRECTION    FOR   SECOND    DIFFERENCES.  215 

we  have,  by  (272), 


Ct  —  C  =  15  a  TJ  -j- 
Ca  —  C  =  15  rt  T3  -f-  flA*m3 
C3  —  C  =  15  a  rs  -f  akm3 
&c.  &c. 


2  sin2  £  r.  2  sin2  Jr.,  «        ,  .      m  ,    Tr 

in  which  Wj  =  —  ;  —  --—  ,  w2  —  —  :  —  —  -,  &c.,  are  found  by  Tab.  V. 
sin  1  sin  1 

with  the  arguments  rl9  T^  &c.     The  mean  of  these  equations, 
observing  that 


gves 


==c        ah  mi 

0 


in  which  n  —  the  number  of  observations.    Or,  denoting  the  mean 
of  the  values  of  m  from  the  table  by  wi0,  that  is,  putting 


we  have 

C  =  C0  —  a*m0  (273) 

Zd.  Reduction  of  the  mean  of  the  times  to  the  mean  of  the  zenith 
distances.  —  Let  TQ  be  the  clock  time  corresponding  to  the  mean 
of  the  zenith  distances,  then  £0  —  £  is  the  change  of  zenith  dis- 
tance in  the  interval  TQ  —  T,  and,  since  this  interval  is  very  small, 
we  shall  have  sensibly 

15  a  (TQ  —  T)  =  ^  —  C  =  akm% 
whence 

Z;=r+A*ro.  (274) 

We  have,  then,  only  to  compute  the  true  time  TQ'  from  the  mean 
of  the  zenith  distances  in  the  usual  manner,  and  the  clock  cor- 
rection will  then  be  found,  as  in  other  cases,  by  the  formula 


To  compute  &,  we  must  either  first  find  q  and  A,  or,  which  is 
preferable,  express  it  by  the  known  quantities.     We  have 

cos  q  cos  A  =  cos  t  —  sin  q  sin  A  cos  C 
sin2  t 

=  COS  t  --       -  COS  <p  COS  d  COS  C 

sin2  C  • 


216 
whence 


TIME. 


sin  t  cos  <f>  cos  8 

sin  C  tan  C 


(275) 


in  which  we  employ  for  £  and  £  the  mean  zenith  distance  and 
the  computed  hour  angle. 

This  mode  of  correction  ig  evidently  more  simple  and  direct 
than  the  first. 

EXAMPLE.— In  St.  Louis,  Lat,  38°  38'  15"  K,  Long.  6A  lm  7'W., 
tne  following  double  altitudes  of  the  sun  were  observed  with  a 
Pistor  and  Martin  prismatic  sextant,  the  index  correction  of 
which  was  +  20".  The  assumed  correction  of  the  chronometer 
to  mean  local  time  was  -f-  2™  12*.  Barom.  30.25  inches,  Att. 
Therm.  80°,  Ext.  Therm.  81°. 


St.: 

Louis,  June  24,  1861. 

2  Q                                                Chronom. 

T 

m 

125° 

15' 

10" 

22*  14™ 

30' 

.5 

6™  42' 

88".  14 

125 

49 

10 

16 

7 

.5 

5      5 

50  .73 

126 

23 

0 

17 

46 

.0 

3    26 

23  .14 

126 

41 

40 

18 

39 

.5 

2    33 

12  .76 

127 

32 

30 

21 

6.5 

0      6 

0  .02 

127 

57 

45 

22 

22 

L 

1    10 

2  .67 

128 

22 

0 

23 

33 

,5 

2    21 

10  .84 

128 

51 

50 

25 

1 

.2 

3    49 

28  .60 

129 

8 

35 

25 

51 

.3 

4    39 

42  .45 

129 

33 

0 

27 

3 

.5 

5    51 

67  .19 

Mean 

127 

33 

28 

T=  22   21 

12 

.15 

7»( 

,  =  32  .65 

4- 

20 

Correction  for  ")  _ 

1 

67 

127 

38 

48 

second  diff.    / 

log  mQ 

1.5139 

Obs'd  Q 
(*)r  « 

63 

46 

54 
27 

.2 

TO  =  22   21 
TQ'  —  22   23 

10.48 
22.94 

los-  is- 

log  cot  t 

8.8239 
nO.3367 

+ 

3 

.7 

&T=  4.     2 

12, 

,46 

—  4*.  7  3 

wO.6745 

P  — 

s  =-. 

4- 

15 

46 

.3 

log  ^5-  ra0 

0.3378 

k0  = 

64 

2 

16 

.8 

log  sin  t 

n9.6215 

£o_ 

25 

57 

43 

.2 

log  cos  0 

9.8927 

<j>  = 

38 

38 

15 

log  cos  6 

9.9627 

6  = 

23 

23 

49 

.3 

log  cosec 

C0    0.3588 

t    —        940  4<v  48" 

4 

log  cot  C0 

0.3125 

—    1* 

38'"  55* 

.23 

—  3'.  06 

nO.4860 

App.   time  = 

22 

21 

4 

.77 

—  1.67 

Eq.  of  time  = 

4- 

2 

18 

.17 

*y= 

22 

23 

22.94 

**The  refraction  should  here  be  the  mean  of  the  refractions  computed  for  the 


CORRECTION    FOR    SECOND    DIFFERENCES.  217 

The  correction  for  second  differences  is  particularly  useful  in 
reducing  series  of  altitudes  observed  with  the  repeating  circle  ;* 
for  with  this  instrument  we  do  not  obtain  the  several  altitudes, 
but  only  their  mean.  (See  Vol.  II.)  When  the  several  altitudes 
are  known,  we  can  avoid  the  correction  by  computing  each 
observation,  or  by  dividing  the  whole  series  into  groups  of  such 
extent  that  within  the  limits  of  each  the  second  differences  will 
be  insensible,  and  computing  the  time  from  the  mean  of  each 
group. 

FOURTH    METHOD. — BY    THE    DISAPPEARANCE    OF   A    STAR    BEHIND   A 
TERRESTRIAL    OBJECT. 

152.  The  rate  of  the  clock  may  be  found  by  this  method  with 
considerable  accuracy  without  the  aid  of  astronomical  instru- 
ments. The  terrestrial  object  should  have  a  sharply  denned 
vertical  edge,  behind  which  the  disappearance  is  to  be  observed, 
and  the  position  of  the  eye  of  the  observer  should  be  precisely 
the  same  at  all  the  observations.  If  the  star's  right  ascension 
and  declination  are  constant,  the  difference  between  the  sidereal 
clock  times  J^  and  T2  of  two  disappearances  is  the  rate  d Tin  the 
interval,  or 

3T=  TI  —  Tz 

but  if  the  right  ascension  a  has  increased  in  the  interval  by  AOC, 
then  the  rate  is 

dT=  T,  —   T2  -j-  Aa 

To  find  the  correction  for  a  small  change  of  declination  =  A#, 

several  altitudes  or  zenith  distances,  but  for  small  zenith  distances  the  difference 
will  be  insensible.  At  great  zenith  distances  we  should  compute  the  several  refrac- 
tions, but  under  80°  we  may  take  the  refraction  r  for  the  mean  apparent  zenith 
distance  z0,  and  correct  it  as  follows :  Take  the  difference  between  z0  and  each  z,  and 
the  mean  m0  of  the  values  of 

m  =  2  sin'  £  (z  -  z0) 
sin  1" 

from  Table  V.  (converting  the  argument  z  —  z0  into  time);  then  the  mean  of  the 
refractions  will  be  found  by  the  formula 

r0  i=  r  -j-  2m0  sin  r  sec2  z0 

The  difference  z  —  z0  should  not  much  exceed  1°. 

*  This  method  was  frequently  practised  in  the  geodetic  survey  of  France.  See 
Nouvelle  Description  Geometrique  de  la  France  (PUISSANT),  Vol.  I.  p.  9G. 


218  TIME. 

we  have,  by  the  second  equation  of  (51),  since  the  azimuth  is  here 
constant  as  well  as  the  latitude,  so  that  dA  =  0  and  dip  =  0, 

A  8  tan  q 

15  C08  d 

and  hence  the  rate  in  the  interval  will  be 

(276) 


. 

15  cos  d 

The  angle  q  will  be  found  with  sufficient  precision  from  an 
approximate  value  of  t  by  (19)  or  (20). 

If  we  know  the  absolute  azimuth  of  the  object,  we  can  find 
the  hour  angle  by  Art.  12,  and  hence  also  the  clock  correction. 

TIME    OF   RISING   AND    SETTING   OF   THE   STARS. 

153.  To  find  the  time  of  true  rising  or  setting,  —  that  is,  the  instant 
when  the  star  is  in  the  true  horizon,  —  we  have  only  to  compute 
the  hour  angle  by  the  formula  (28)  :  \^* 

cos  t  =  —  tan  <p  tan  d 
and  then  deduce  the  local  time  by  Art.  55. 

154.  To  find  the  time  of  apparent  rising  or  setting  -,  —  that  is,  the 
instant  when  the  star  appears  on  the  horizon  of  the  observer,  —  we 
must  allow  for  the  horizontal  refraction.    Denoting  this  refraction 
by  r0,  the  true  zenith  distance  of  the  star  at  the  time  of  apparent 
rising  or  setting  is  90°  -j-  r0,  and,  employing  this  value  for  £,  we 
compute  the  hour  angle  by  (267). 

Since  the  altitude  h  =  90°  —  f  ,  we  have  in  this  case  h  =  —  r0, 
with  which  we  can  compute  the  hour  angle  by  the  formula  (268). 

In  common  life,  by  the  time  of  sunrise  or  sunset  is  meant  the 
instant  when  the  sun's  upper  limb  appears  in  the  horizon.  The 
true  zenith  distance  of  the  centre  is,  then,  £  =  90°  -j-  r0  —  x  +  S 
(where  TT  =  the  horizontal  parallax  and  S  —  the  semidiameter), 
with  which  we  compute  the  hour  angle  as  before.  The  same 
form  is  to  be  used  for  the  moon. 

TIME   OF  THE   BEGINNING  AND   ENDING   OF  TWILIGHT. 

155.  Twilight  begins  in  the  morning  or  ends  in  the  evening 
when  the  sun  is  18°  below  the  horizon,  and  consequently  the 


AT    SEA.  219 

zenith  distance  is  then  £  =  90°  +  18°,  or  h  =  —  18°,  with  which 
we  can  find  the  hour  angle  by  (267)  or  (268). 

NOTE. — Methods  of  finding  at  once  both  the  time  and  the  latitude  from  observed 
altitudes  will  be  treated  of  under  Latitude,  in  the  next  chapter. 

FINDING   THE    TIME   AT    SEA. 

First  Method. — By  a  Single  Altitude. 

156.  This  is  the  most  common  method  among  navigators,  as 
altitudes  from  the  sea  horizon  are  observed  with  the  greatest 
facility  with  the  sextant.  Denoting  the  observed  altitude  cor- 
rected for  the  index  error  of  the  sextant  by  H,  the  dip  of  the 
horizon  by  .D,  Ave  have  the  apparent  altitude  h'  =  II  —  D;  then, 
taking  the  refraction  r  for  the  argument  A',  the  true  altitude  of  a 
star  is  h  =  h'  —  r.  A  planet  is  observed  by  bringing  the  esti- 
mated centre  of  its  reflected  image  upon  the  horizon,  so  that  no 
correction  for  the  semidiameter  is  employed;  the  parallax  is  com- 
puted by  the  simple  formula  (TT  being  the  horizontal  parallax) 

p  —  TT  cos  h' 
and  hence  for  a  planet 

h  =  h'  —  r  +  r  cos  ft 

The  moon  and  sun  are  observed  by  bringing  the  reflected 
image  of  either  the  upper  or  the  lower  limb  to  touch  the  horizon. 
As  very  great  precision  is  neither  possible  nor  necessary  in  these 
observations,  the  compression  of  the  earth  is  neglected,  and  the 
parallax  is  computed  by  the  formula 

p  =  TT  cos  (hf  —  r) 
and  then,  S  being  the  semidiameter, 

h  ==  V  —  r  -f  TT  cos  (h'  —  r)  ±  S 

In  nautical  works,  the  whole  correction  of  the  moon's  altitude 
for  parallax  and  refraction  =  TT  cos  (h!  —  r)  —  r  is  given  in  a  table 
with  the  arguments  apparent  altitude  (h'}  and  horizontal  parallax 
(TT).  In  the  construction  of  this  table  the  mean  refraction  is  used, 
but  the  corrections  for  the  barometer  and  thermometer  are  given 
in  a  very  simple  table,  although  they  are  not  usually  of  sufficient 
importance  to  he  regarded  in  correcting  altitudes  of  the  moon 
which  are  taken  to  determine  the  local  time. 


220  TIME. 

The  hour  angle  is  usually  found  by  (268). 

It  is  important  at  sea,  where  the  latitude  is  always  in  some 
degree  uncertain,  to  find  the  time  by  altitudes  near  the  prime 
vertical,  where  the  error  of  latitude  has  little  or  no  effect 
(Art.  149). 

157.  The  instant  when  the  sun's  limb  touches  the  sea  horizon 
may  be  observed,  instead  of  measuring  an  altitude  with  the  sex- 
tant.    In  this  case  the  refraction  should  be  taken  for  the  zenith 
distance  90°  -f-  .D,  but,  on  account  of  the  uncertainty  in  the  hori- 
zontal refraction,  great  precision  is  not  to  be  expected,  and  the 
mean   horizontal   refraction  ?*0  may  be   used.     We   then   have 
r  =  90°  -f  D  +  r0  —  *  ±  S,  with  which  we  proceed  by  (267).    In 
so  rude  a  method,  n  may  be  neglected,  and  we  may  take  16'  as 
the  mean  value  of  $,  36'  as  the  value  of  r0,  4'  as  the  average 
value  of  D  from  the  deck  of  most  vessels ;  then  for  the  lower 
limb  we  have  £  =  909  56',  and  for  the  upper  limb  £  =  90°  24'.     If 
both  limbs  have  been  observed  and  the  mean  of  the  times  is 
taken,  the  corresponding  hour  angle  will  be  found  by  taking 
£  =  90°  40'. 

Second  Method. — By  Equal  Altitudes. 

158.  The  method  of  equal  altitudes  as  explained  in  Arts.  139 
and  140  may  be  applied  at  sea  by  introducing  a  correction  for 
the  ship's  change  of  place  between  the  two  observations.     If, 
however,  the  ship  sails  due  east  or  west  between  the  observa- 
tions, and  thus  without  changing  her  latitude,  no  correction  for 
her  change  of  place  is  necessary,  for  the  middle  time  will  evi- 
dently correspond  to  the  instant  of  transit  of  the  star  over  the 
middle  meridian  between  the  two  meridians  on  which  the  equal 
altitudes  are  observed.     But,  if  the  ship  changes  her  latitude, 
let 

A^>  =  the  increase  of  latitude  at  the  second  observation; 
then  (Art.  149)  the  effect  upon  the  second  hour  angle  is 

A> 


15  cos  <p  tan  A 

which  is  the  correction  subtract! ve  from  the  second  observed 
time  to  reduce  it  to  that  which  would  have  been  observed  if  the 


AT    SEA.  221 

ship  had  not  changed  her  latitude  or  had  run  upon  a  parallel. 
Hence  }  &t  is  to  be  subtracted  from  the  mean  of  the  chrono- 
meter times  to  obtain  the  chronometer  time  of  the  star's  transit 
over  the  middle  meridian. 

In  this  formula  we  must  observe  the  sign  of  tan  A.  It  will 
be  more  convenient  in  practice  to  disregard  the  signs,  and  to 
apply  the  numerical  value  of  the  correction  to  the  middle  time 
according  to  the  following  simple  rule  :  —  add  the  correction  when 
the  ship  has  receded  from  the  sun  ;  subtract  it  when  the  ship  has 
approached  the  sun. 

The  azimuth  may  be  found  by  the  formula 

sin  t  cos  S 

sin  A  =  - 

cos  h 

in  which  for  t  we  take  one-half  the  elapsed  time. 

The  sun  being  the  only  object  which  is  employed  in  this  way, 
we  should  also  apply  the  equation  of  equal  altitudes,  Art.  140: 
but,  as  the  greatest  change  of  the  sun's  declination  in  one  hour 
is  about  1',  and  the  change  of  the  ship's  latitude  is  generally 
much  greater,  the  equation  is  commonly  neglected  as  relatively 
unimportant  in  a  method  which  at  sea  is  necessarily  but  ap- 
proximate. But,  if  required,  the  equation  may  be  computed 
and  applied  precisely  as  if  the  ship  had  been  at  rest. 

EXAMPLE.  —  At  sea,  March  20,  1856,  the  latitude  at  noon  being 
39°  N.,  the  same  altitude  was  observed  A.M.  and  P.M.  as  fol- 
lows, by  a  chronometer  regulated  to  mean  Greenwich  time  : 


30°    0'  A.M.  Chro.  time   =  11*  39-  33 

Index  corr.          —    2  P.M.       "         "       =    6   20    17 

Dip  —    4  Elapsed  time  =  2t=    6   40    44 

Refraction  2  Middle  time  =    2   59    55 

Semidiam.  4-  16  Chron.  correction  =     —  2    12 


h  =  30     8          Green,   time    of 


noon 


}=    2   57    43 


The  ship  changed  her  latitude  between  the  two  observations 
by  AC*  =  -20'=  -1200".  For  the  Greenwich  date  March 
20,  2*  58W,  the  Ephemeris  gives  d  =  +  0°  4',  and  we  have  t  = 
3*  20-  22«  =  50°  5'  30",  <p  =  39°  0'.  Hence 


222  TIME. 

log  sin  t    9.8848  log  ^  8.5229 

log  cos  <5  0.0000  log  AC*  3.0792 

log  sec  h  0.0631  log  sec  <p  0.1095 

log  sin  A  9.9479  log  cot  A  9.7165 

log  26'.8  1.4281 

The  ship  has  approached  the  sun,  and  hence  268.8  must  be  sub- 
tracted from  the  middle  time. 

If  we  wish  to  apply  the  equation  of  equal  altitudes,  we  have 
further  from  the  Ephemeris  &'d  =  -f-  59",  and  hence,  by  Art. 

140, 

log  A       n9.4628  log  B  9.2698 

log  A'<5       1.7709  log  A'<S  1.7709 

log  tan  <f   9.9084  log  tan  d  7.0658 

a  =  —  13'.9  log  a   rcl.1421  b  =  -f  O'.O  log  b  8.1065 

Hence  we  have 

Chro.  middle  time  =  2*  59"1  55'. 

Corr.  for  change  of  lat.  =  26  .8 

Equation  of  eq.  alts.        =       —    13  .9 
Chro.  time  app.  noon       =2   59    14 .3 

At  sea,  instead  of  using  the  observation  to  iind  the  chrono- 
meter correction,  we  use  it  to  determine  the  ship's  longitude  (as 
will  be  fully  shown  hereafter) ;  and  therefore,  to  carry  the  opera- 
tion out  to  the  end,  we  shall  have 

Chro.  time  app.  noon  =    2*  59m  14* 

Corr.  of  chronom.  =j-       2    12 

Green,  mean  time  noon  =    2   57      2 

Equation  of  time  7    48 

Greenwich  app.  time  at  the  local  noon  =    2   49    14 

which  is  the  longitude  of  the  middle  meridian,  or  the  longitude 

of  the  ship  at  noon. 

* 

159.  In  low  latitudes  (as  within  the  tropics)  observations  for 
the  time  may  be  taken  when  the  sun  is  very  near  the  meridian, 
for  the  condition  that  the  sun  should  be  near  the  prime  vertical 
may  then  be  satisfied  within  a  few  minutes  of  noon ;  and  in  case 
the  ship's  latitude  is  exactly  equal  to  the  declination,  it  will  be 
satisfied  only  when  the  sun  is  on  the  meridian  in  the  zenith.  In 
such  cases  the  two  equal  altitudes  may  be  observed  within  a  few 
minutes  of  each  other,  and  all  corrections,  whether  for  change 
of  latitude  or  change  of  declination,  may  be  disregarded. 


MERIDIAN    ALTITUDES.  223 


CHAPTER  VI. 

FINDING   THE    LATITUDE   BY   ASTRONOMICAL   OBSERVATIONS. 

160.  BY  the  definition,  Art.  7,  the  latitude  of  a  place  on  the 
surface  of  the  earth  is  the  declination  of  the  zenith.     It  was  also 
shown  in  Art.  8  to  be  equal  to  the  altitude  of  the  north  pole  above 
the  horizon  of  the  place.     In  adopting  the  latter  definition,  it  is 
to  be  remembered  that  a  depression  below  the  horizon  is  a 
negative   altitude,   and   that   south   latitude   is   negative.     The 
south  latitude  of  a  place,  considered  numerically,  or  without 
regard  to  its  algebraic  sign,  is  equal  to  the  elevation  of  the 
south  pole. 

It  is  to  be  remembered,  also,  that  the  latitude  thus  defined  is 
not  an  angle  at  the  centre  of  the  earth  measured  by  an  arc  of 
the  meridian,  as  it  would  be  if  the  earth  were  a  sphere  ;  but  it 
is  the  angle  which  the  vertical  line  at  the  place  makes  with  the 
plane  of  the  equator,  Art.  81. 

We  have  seen,  Art.  86,  that  there  are  abnormal  deviations  of 
the  plumb  line,  which  make  it  necessary  to  distinguish  between 
the  geodetic  and  the  astronomical  latitude.  We  shall  here  treat  ex- 
clusively of  the  methods  of  determining  the  astronomical  lati- 
tude ;  for  this  depends  only  upon  the  actual  position  of  the 
plumb  line,  and  is  merely  the  declination  of  that  point  of  the 
heavens  towards  which  the  plumb  line  is  directed. 

FIRST    METHOD. — BY    MERIDIAN    ALTITUDES    OR    ZENITH    DISTANCES. 

161.  Let  the  altitude  or  zenith  distance  of  a  star  of  known 
declination  be  observed  at  the  instant  when  it  is  on  the  meridian. 
Deduce  the  true  geocentric  zenith  distance  £,  and  let  d  be  the 
geocentric  declination,  <p  the  astronomical  latitude. 

Let  the  celestial  sphere  be  projected  on  the  plane  of  the 
meridian,  and  let  ZNZ',  Fig.  24,  be  the  celestial  meridian;  0 
the  centre  of  the  sphere  coincident  with  that  of  the  earth;  POP' 
the  axis  of  the  sphere;  P  the  north  pole;  and  ECQihe  projection 


224 


LATITUDE. 


of  the  plane  of  the  equinoctial.  Let  CZ  be  parallel  to  the 
vertical  line  of  the  observer;  then  the  point  Z  of  the  celestial 
sphere,  being  the  vanishing  point  of  all 
lines  parallel  to  CZ,  is  the  astronomical 
zenith  of  the  observer,  and  ZE=^  the  astro- 
nomical latitude  —  <p.  If,  then,  A  is  the 
position  of  the  star  on  the  meridian,  north 
of  the  equator  but  south  of  the  zenith,  we 
have  ZA  =  £,  AE  =  8,  and  hence 


<?=  <*  +  C 


(277) 


This  equation  may  be  treated  as  entirely  general  by  attending 
to  the  signs  of  8  and  f .  Since  in  deducing  it  we  supposed  the 
star  to  be  north  of  the  equator,  it  holds  for  the  case  where  it  is 
south  by  giving  the  declination  in  that  case  the  negative  sign, 
according  to  the  established  practice ;  and,  since  we  supposed 
the  star  to  be  south  of  the  zenith,  the  equation  will  hold  for  the 
case  where  it  is  north  of  the  zenith  by  giving  £  in  that  case  the 
negative  sign.  If  the  star  is  so  far  north  of  the  zenith  as  to  be 
below  the  pole,  or  at  its  lower  culmination,  the  equation  will 
still  hold,  provided  we  still  understand  by  o  the  star's  distance 
north  of  the  equator,  measured  from  E  through  the  zenith  and 
elevated  pole,  or  the  arc  EAf.  This  arc  is  the  supplement  of  the 
declination ;  and  we  may  here  remark  that,  in  general,  any 
formula  deduced  for  the  case  of  a  star  above  the  pole  will 
apply  to  the  case  where  it'  is  below  the  pole  by  employing  the 
supplement  of  the  declination  instead  of  the  declination  itself; 
that  is,  by  reckoning  the  declination  over  the  pole. 

The  case  of  a  star  below  the  pole  is,  however,  usually  con- 
sidered under  the  following  simple  form.  Put 


then 


P  =  PA'  =  the  star's  polar  distance, 
h  =  NA'  =  "       true  altitude, 


(278) 


in  which  for  south  latitude  P  must  be  the  star's  south  polar  dis- 
tance, and  the  sum  of  P  and  h  is  only  the  numerical  value  of  <p. 

The  declination  is  to  be  found  for  the  instant  of  the  meridian 
transit  by  Art.  GO  or  G2. 

In   the  observatory,  instruments   are  employed   which   give 


MERIDIAN   ALTITUDES.  225 

directly  the  zenith  distance,  or  its  supplement,  the  nadir  distance. 
With  a  meridian  circle  perfectly  adjusted  in  the  meridian,  the 
instant  of  transit  would  he  known  without  reference  to  the 
clock,  and  the  observation  would  be  made  at  the  instant  the 
star  passed  the  middle  thread  of  the  reticule  ;  hut  when  the  in- 
strument is  not  exactly  in  the  meridian,  or  when  the  observation 
is  not  made  on  the  middle  thread,  the  observed  zenith  distance 
must  be  reduced  to  the  meridian,  for  which  see  Vol.  II.,  Meridian 
Circle. 

With  the  sextant  or  other  portable  instruments  the  meridian 
altitude  of  a  fixed  star  may  be  distinguished  as  the  greatest 
altitude,  and  no  reference  to  the  time  is  necessary.  But,  as  the 
sun,  moon,  and  planets  constantly  change  their  declination, 
their  greatest  altitudes  may  be  reached  either  before  or  after  the 
meridian  passage  ;*  and  in  order  to  observe  a  strictly  meridian 
altitude  the  clock  time  of  transit  must  be  previously  computed 
and  the  altitude  observed  at  that  time. 

EXAMPLE  1.— On  March  1, 1856,  in  Long.  I0h  5™  32*  E.,  suppose 
the  apparent  meridian  altitude  of  the  sun's  lower  limb,  north  of 
the  zenith,  is  63°  49'  50",  Barom.  30.  in.,  Ext.  Therm.  50°;  what 
is  the  latitude  ? 


App.  zen.  dist. 

r   — 

P  = 

S  = 

C  = 

d  = 

9  = 

26° 

+ 

10' 
16 

10 

28 
3 
10 

tt 

.7 

.8 
.3 

—  26 

7 

—  33~ 

26 
33 

59 

45 
5 
51 

.2 

.8 
~JO 

EXAMPLE  2. — July  20,  1856,  suppose  that  at  a  certain  place 
the  true  zenith  distances  of  a  Aquilce,  south  of  the  zenith,  and 
a  Cephei  north  of  the  zenith,  have  been  obtained  as  follows : 

a  Aquilse  +  a  Cephei 

C  =  +  26°  34'  27".5  C  =  —  26°  54'  28".3 

d  ==  +    8    29  22  .7  d  =  +  61    58  21  .1 

<p  =  -f  35      3  50  .2  ?  =  -f  35  ~3.  52  .8 

The   mean   latitude   obtained   by  the   two   stars   is,  therefore, 
<p  ==  -f-  35°  3'  51". 5.     In  this  example,  the  stars  being  at  nearly 

*  See  Art.  172  for  the  method  of  finding  the  time  of  the  sun's  greatest  altitude, 
which  may  also  be  used  for  the  moon  or  a  planet. 
VOL.  I.— 15 


226 


LATITUDE. 


the  same  zenith  distance,  but  on  opposite  sides  of  the  zenith,  any 
constant  though  unknown  error  of  the  instrument,  peculiar  to 
that  zenith  distance,  is  eliminated  in  taking  the  mean.  Thus, 
if  the  zenith  distance  in  both  cases  had  been  10"  greater,  we 
should  have  found  from  a  Aquilce  <p  =  35°  4'  0".2,  but  from 
a  Cephei  <p  =  35°  3'  42".8,  but  the  mean  would  still  be  <p  =  35°  3' 
51".5. 

It  is  evident,  also,  that  errors  in  the  refraction,  whether  due  to 
the  tables  or  to  constant  errors  of  the  barometer  'and  thermo- 
meter, or  to  any  peculiar  state  of  the  air  common  to  the  two 
observations,  are  nearly  or  quite  eliminated  by  thus  combining  a 
pair  of  stars  the  mean  of  whose  declinations  is  nearly  equal  to 
the  declination  of  the  zenith.  The  advantages  of  such  a  com- 
bination do  not  end  here.  If  we  select  the  two  stars  so  that  the 
difference  of  their  zenith  distances  is  so  small  that  it  may  be 
measured  with  a  micrometer  attached  to  a  telescope  which  is  so 
mounted  that  it  may  be  successively  directed  upon  the  two  stars 
without  disturbing  the  angle  which  it  makes  with  the  vertical 
line,  we  can  dispense  altogether  with  a  graduated  circle,  or,  at 
least,  the  result  obtained  will  be  altogether  independent  of  its 
indications.  For,  let  £  and  £'  be  the  zenith  distances,  d  and  of 
the  declinations  of  the  two  stars,  the  second  of  which  is  north  of 
the  zenith  ;  then,  if  f  denotes  only  the  numerical  value  of  the 
zenith  distance,  we  have 


the  mean  of  which  is 


(279) 


so  that  the  result  depends  only  upon  the  given  declinations  and 
the  observed  difference  of  zenith  distance  which  is  measured  with 
the  micrometer.  Such  is  the  simple  principle  of  the  method  first 
introduced  by  Captain  TALCOTT,  and  now  extensively  used  in  this 
country.  To  give  it  full  effect,  {he  instrument  formerly  known 
as  the  Zenith  Telescope  in  England  has  received  several  important 
modifications  from  our  Coast  Survey.  It  will  be  fully  treated  of, 
in  its  present  improved  form,  in  Vol.  II.,  where  also  will  be 
found  a  discussion  of  TALCOTT'S  method  in  all  its  details.  • 

162.  Meridian  altitudes  of  a  circumpolar  star  observed  both  above 
and  beloio  the  pole.  —  Every  star  whose  distance  from  the  elevated 


MERIDIAN   ALTITUDES.  227 

pole  is  less  than  the  latitude  may  be  observed  at  both  its  upper 
and  lower  culminations.  If  we  put 

h  —  the  true  altitude  at  the  upper  culmination, 

Aj  =       "  "  "        lower  " 

p  =  the  star's  polar  distance  at  the  upper  culmination, 

pl  =       "  "  "  "        lower  " 

we  have,  evidently, 

<p  =  h  —  p 

<P  —  ^i  H-  Pi 
the  mean  of  which  is 

<p  =  J  (h  -f-  7ij)  -f  i  (pl  —  p')  (280) 

whence  it  appears  that  by  this  method  the  absolute  values  of  p 
and^j  are  not  required,  but  only  their  difference  pl — p.  The 
change  of  a  star's  declination  by  precession  and  nutation  is  so 
small  in  12*  as  usually  to  be  neglected,  but  for  extreme  precision 
ought  to  be  allowed  for.  This  method,  then,  is  free  from  any 
error  in  the  declination  of  the  star,  and  is,  therefore,  employed 
in  all  fixed  observatories. 

EXAMPLE. — With  the  meridian  circle  of  the  N"aval  Academy 
the  upper  and  lower  transits  of  Polaris  were  observed  in  1853 
Sept.  15  and  16,  and  the  altitudes  deduced  were  as  below:  *- 

Upper  Transit.  Lower  Transit. 

Sept.  15,  App.  alt.  40°28'25".42  Sept.  16,  37°31'39".76 

Barom.       30.005    ^  Barom.         30.146^ 

Att.  Therm.  65°. 2   I  Ref.          1     6.34  Att.  Therm.  75°      [Ref.  112.45 

Ext.    «         63  .8  J  Ext.    «          74  .eJ 

h  —  40  27  19  .08  ^  =  37  30  27  .31 

p—    1   28  2G  .04  p,  =    1   28  25  .86 


^  =  38  58  53  .04  0  =  38  58  53  .17 

"  "  53  .04 
Mean  <t>  =  38  58  53  .11 

In  order  to  compare  the  results,  each  observation  is  carried 
out  separately.     By  (280)  we  should  have 

}  (h   -f  fc,)  =  38°  58'  53".20 

^(P1-P)= -     0  -09 

9  =  38    58  53  .11 

This  method  is  still  subject  to  the  whole  error  in  the  refraction, 


228  LATITUDE. 

which,  however,  in  the  present  state  of  the  tables,  will  usually  be 
very  small. 

If  the  latitude  is  greater  than  45°,  and  the  star's  declination 
less  than  45°,  the  upper  transit  occurs  on  the  opposite  side  of  the 
zenith  from  the  pole.  In  that  case  h  must  still  represent  the 
distance  of  the  star  from  the  point  of  the  horizon  below  the  pole, 
and  will  exceed  90°.  Thus,  among  the  Greenwich  observations 
we  find  v 

1837  June  14,  Capella  7^  =  7°  18'  7".94 
h  =  95  39  7  .91 
?  =  51  28  37  .93 

163.  Meridian  zenith  distances  of  the  sun  observed  near  the  summer 
and  winter  solstices.  —  When  the  place  of  observation  is  near  the 
equator,  the  lower  culminations  of  stars  can  no  longer  be  ob- 
served, and,  consequently,  the  method  of  the  preceding  article 
cannot  be  used.  The  latitude  found  from  stars  observed  at  their 
upper  culminations  only  is  dependent  upon  the  tabular  declina- 
tion, and  is,  therefore,  subject  to  the  error  of  this  declination.  If, 
therefore,  an  observatory  is  established  on  or  near  the  equator, 
and  its  latitude  is  to  be  fixed  independently  of  observations  made 
at  other  places,  the  meridian  zenith  distances  of  stars  cannot  be 
employed.  The  only  independent  method  is  then  by  meridian 
observations  of  the  sun  near  the  solstices. 

Let  us  at  first  suppose  that  the  observations  can  be  obtained 
exactly  at  the  solstice,  and  the  obliquity  (s)  of  the  ecliptic  is 
constant.  The  declination  of  the  sun  at  the  summer  solstice  is 
S  -f-  e,  and  at  the  winter  solstice  it  is  =  —  s  ;  hence,  from  the 
meridian  zenith  distances  £  and  £'  observed  at  these  times,  we 
should  have 

9  =  :  -f  * 

p  =  C'*M  * 

the  mean  of  which  is 


a  result  dependent  only  upon  the  data  furnished  by  the  observa- 
tions. 

Now,  the  sun  will  not,  in  general,  pass  the  meridian  of  the 
observer  at  the  instant  of  the  solstice,  or  when  the  declination  is 
at  its  maximum  value  e;  nor  is  the  obliquity  of  the  ecliptic  con- 
stant. But  the  changes  of  the  declination  near  the  solstices  are 
very  small,  and  hence  are  very  accurately  obtained  from  the 


ALTITUDE    AT    A    GIVEN    TIME.  229 

solar  tables  (or  from  the  Ephemeris  which  is  based  on  these 
tables),  notwithstanding  small  errors  in  the  absolute  value  of  the 
obliquity.  The  small  change  in  the  obliquity  between  two 
solstices  is  also  very  accurately  known.  If  then  AS  is  the  un- 
known correction  of  the  tabular  obliquity,  and  the  tabular  values 
at  the  two  solstices  are  e  and  s',  the  true  values  are  £  -f-  AS  and 
£f  +  AS  ;  and  if  the  tabular  declinations  at  two  observations  near 
the  solstices  are  e  —  x  and  —  (s'  —  xr),  the  true  declinations  will 
be  3  =  s  -f  AS  —  x  and  df  =  —  (e'  -f-  AS  —  x'},  and  by  the  formula 
<p  —  £  +  o  we  shall  have  for  the  two  observations 

<p  =  C    -}-  s    -f  As  —  X 

<?  =  :'  —  £'  —  AC  -fa/ 

the  mean  of  which  is 

V  =  *  (C  +  C')  +  J  (e  -  e')  -  J  O  -  x') 

a  result  which  depends  upon  the  small  changes  s  —  sr  and  x  —  x', 
both  of  which  are  accurately  known. 

It  is  plain  that,  instead  of  computing  these  changes  directly,  it 
suffices  to  deduce  the  latitude  from  a  number  of  observations 
near  each  solstice  by  employing  the  apparent  declinations  of  the 
solar  tables  or  the  Ephemeris  ;  then,  if  <p'  is  the  mean  value  of 
the  latitude  found  from  all  the  observations  at  the  northern 
solstice,  and  <p"  the  mean  from  all  at  the  southern  solstice,  the 
true  latitude  will  be 


Every  observation  should  be  the  mean  of  the  observed  zenith 
distances  of  both  the  upper  and  the  lower  limb  of  the  sun,  in 
order  to  be  independent  of  the  tabular  semidiameter  and  to 
eliminate  errors  of  observation  as  far  as  possible. 

SECOND    METHOD.  -  BY    A    SINGLE    ALTITUDE    AT    A    GIVEN    TIME. 

164.  At  the  instant  when  the  altitude  is  observed,  the  time  is 
noted  by  the  clock.  The  clock  correction  being  known,  we  find 
the  true  local  time,  and  hence  the  star's  hour  angle,  by  the 
formula 

t  =  e  —  a 

in  which  0  is  the  sidereal  time  and  a  the  star's  riijlit  ascension. 


230  LATITUDE. 

If  the  sun  is  observed,  t  is  simply  the  apparent  solar  time.     We 
have,  then,  by  the  first  equation  of  (14), 

sin  <f>  sin  d  -j-  cos  <p  cos  d  cos  t  =  sin  h 

in  which  <p  is  the  only  unknown  quantity.     To  determine   it, 
assume  d  and  D  to  satisfy  the  conditions 

d  sin  D  =  sin  S 
d  cos  D  =  cos  d  cos  t 
then  the  above  equation  becomes 

d  cos  (^  —  D)  —  sin  A 

which  determines  tp  —  Z>,  and  hence  also  <p.      For  practical  con- 
venience, however,  put 


then,  by  eliminating  d,  the  solution  may  be  put  under  the  follow- 
tan  D  =  tan  d  sec  t 


ing  form: 


cos    f  =  sin  A  sin  D  cosec 
=  D  ± 


(281) 


The  first  of  these  equations  fully  determines  JD,  which  will  be 
taken  numerically  less  than  90°,  positive  or  negative  according 
to  the  sign  of  its  tangent.  As  t  should  always  be  less  than  90°, 
or  67',  D  will  have  the  same  sign  as  d. 

The  second  equation  is  indeterminate  as  to  the  sign  of  f, 
since  the  cosine  of  +  f  and  —  r  are  the  same.  Hence  we 
obtain  by  the  third  equation  two  values  of  the  latitude.  Only 
one  of  these  values,  however,  is  admissible  wheji  the  other  is 
numerically  greater  than  90°,  which  is  the  maximum  limit  of 
latitudes.  When  both  values  are  within  the  limits  -f  90°  and 
-  90°,  the  true  solution  is  to  be  distinguished  as  that  which 
agrees  best  with  the  approximate  latitude,  which  is  always  suffi- 
ciently well  known  for  this  purpose,  except  in  some  peculiar 
cases  at  sea. 

EXAMPLE  1.— 1856  March  27,  in  the  assumed  latitude  23°  8. 
and  longitude  43°  14'  "W.,  the  double  altitude  of  the  sun's  lower 


ALTITUDE    AT    A   GIVEX    TIME. 


231 


limb  observed  with  the  sextant  and  artificial  horizon  was  114° 
40'  30"  at  4A  21'"  15"  by  a  Greenwich  Chronometer,  which  wab 
fast  2m  30".  Index  Correction  of  Sextant  =  -  V  12",  Barom. 
29.72  inches,  Att.  Therm.  61°  F.,  Ext.  Therm.  61°  F.  Required 
the  true  latitude. 


Sextant  reading  *=  114°  40'  30" 
Index  corr.  1   12 

114    39  18~ 


Chronometer 
Correction 


4*  21-  15* 
—     2   30 


App.  alt.  Q 
Semidiameter      = 
Ref.  and  par.        = 

57  19  39 
+  16  3 
—  31 

h  = 

57  35  11 

Gr.  date,  March  27,  4  18   45 
Longitude          =     2  52   56 
Local  mean  t. 
Eq.  of  time 
App.  time,  t 


=     1  25 

=  —     5 


49 
19 


d  =  -|-2    51  30 


log  sec  t 
log  tan  d 


0.027360 
8.698351 


log  tan  D     8.725711 


r  = 
D  —  r==  <?  = 


3°  2'  38" 
25  58  49 
22  56  11 


=     1  20   30 
=  20°   T  30" 


log  cosec  8  1.302190 

log  sin  D  8.725098 

log  sin  h  9.926445 

log  cos  f  9.953733 


EXAMPLE  2. — 1856  Aug.  22;  suppose  the  true  altitude  of 
Fomalhaut  is  found  to  be  29°  10'  0"  when  the  local  sidereal  time 
is  21A  49W  44" ;  what  is  the  latitude  ? 

We  have  a  =  22* 49-  44*,  whence  f  =  —  1*  0-  O'j  <J  =  —  80°  22'  47".5  ; 
D=  —  31°  15'  13",  r  =  ±  60°  0'  6",  <p  =-.  -f  28°  44'  53".  The  nega- 
tive value  of  f  here  gives  <p  =  —  91°  15'  19";  which  is  inadmissible. 

165.  The  observation  of  equal  altitudes  east  and  west  of  the 
meridian  may  be  used  not  only  for  determining  the  time  (Art. 
139),  but  also  the  latitude.  For  the  half  elapsed  sidereal  time 
between  two  such  altitudes  of  a  tixed  star  is  at  once  the  hour 
angle  required  in  the  method  of  the  preceding  article.  When 
the  sun  is  used  in  this  way,  the  half  difference  between  the 
apparent  times  of  the  observations  is  the  hour  angle,  and  the 
declination  must  be  taken  for  noon,  or  more  strictly  for  the 
mean  of  the  times  of  observation.  By  thus  employing  the 
mean  of  the  A.M.  and  P.M.  hour  angles  and  the  mean  of  the 
corresponding  declinations,  we  obtain  sensibly  the  same  result 


232  LATITUDE. 

as  by  computing  each  observation  separately  with  its  proper 
hour  angle  and  declination  and  then  taking  the  mean  of  the 
two  resulting  latitudes ;  and  an  error  in  the  clock  correction 
does  not  affect  the  final  result.  The  clock  rate,  however,  must 
be  known,  as  it  affects  the  elapsed  interval.  See  also  Art.  182. 

166.  Effect  of  errors  in  the  data  upon  the  latitude  computed  from  an 
observed  altitude. — Erom  the  first  of  the  equations  (51)  we  find 

d*          sin  q  cos<5  cos  q 

dd 


or,  since  h  =  90°  —  £,  dh  =  —  c/£,  and  sin  q  cos  S  =  cos  <p  sin  A, 
d(f>  =  —  sec  A.dh  —  cos  <p  tan  A .  dt  -j-  cos  q  sec  A .  dd 

whence  it  appears  that  errors  of  altitude  and  time  will  have  the 
least  effect  when  A  —  0  or  180°,  that  is,  when  the  observation  is 
in  the  meridian,  and  the  greatest  effect  when  the  observation  is 
on  the  prime  vertical.  If  the  same  star  is  observed  on  both 
sides  of  the  meridian  and  at  equal  distances  from  it,  the  coeffi- 
cient of  dt  will  have  opposite  signs  at  the  two  observations,  and 
hence  a  small  error  in  the  time  will  be  wholly  eliminated  by 
taking  the  mean  of  the  values  of  the  latitude,  found  from  two 
such  observations.  It  is  advisable,  therefore,  in  taking  a  series 
of  observations,  to  distribute  them  symmetrically  with  respect  to 
the  meridian.  When  they  are  all  taken  very  near  to  the  meri- 
dian, a  special  method  of  reduction  is  used,  which  will  be 
treated  of  below  as  our  Third  Method  of  finding  the  latitude. 

The  sign  of  sec  A  is  different  for  stars  north  and  south  of 
the  zenith :  hence  errors  of  altitude  will  be  at  least  partially 
eliminated  by  taking  the  mean  of  the  results  found  from  stars 
near  the  meridian,  both  north  and  south  of  the  zenith.  A 
constant  error  of  the  instrument  may  thus  be  whotty  eliminated. 

As  for  the  effect  of  the  error  do,  its  coefficient  is  zero  only 
when  q  =  90°  and  sec  A  is  not  infinite.  This  occurs  when  a 
circumpolar  star  is  observed  at  its  elongation,  where  we  have. 
Art.  18, 

COS  (f> 

sec  A  = 


j/[sin  (d  -f  y)  sin  (d  —  ?)] 

which  shows  that  sec  A  diminishes  as  d  increases.     In  order, 
therefore,  to  reduce  the  effect  of  an  error  in  the  declination 


REDUCTION    TO    THE    MERIDIAN.  233 

at  the  same  time  with  that  of  errors  of  altitude  and  time,  we 
should  select  a  star  as  near  the  pole  as  possible,  and  observe  it 
at  or  near  its  greatest  elongation,  011  cither  side  of  the  meridian. 
The  proximity  of  the  star  to  the  pole  enables  us  to  faeilitate  the 
reduction  of  a  series  of  observations,  and  we  shall  therefore 
treat  specially  of  this  case  as  our  Fourth  Method  below. 

167.  When  several  altitudes  not  very  far  from  the  meridian  are 
observed  in  succession,  if  we  wish  to  use  their  mean  as  a  single 
altitude,  the  correction  for  second  differences  (Art.  151)  must  be 
applied.     It  is,  however,  preferable  to  incur  the  labor  of  a  sepa- 
rate reduction  of  each  altitude,  as  we  shall  then  be  able  to  com- 
pare the  several  results,  and  to  discuss  the  probable  errors  of  the 
observations  and  of  the  final  mean.     "When  the  observations  are 
very  near  to  the  meridian,   this  separate  reduction  is  readily 
effected,  with  but  little  additional  labor,  by  the  following  method: 

THIRD    METHOD. BY    REDUCTION    TO    THE    MERIDIAN    WHEN    THE 

TIME    IS    GIVEN. 

168.  To  reduce  an  altitude,  observed  at  a  given  time,  to  the  meridian. — 
This  is  done  in  various  ways. 

(A.)  If  in  the  formula,  employed  in  Art.  164, 

sin  <p  sin  d  -f-  cos  y  cos  8  cos  t  =  sin  h 
we  substitute 

cos  t  =  1  —  2  sin2  $  t 
it  becomes 

sin  <p  sin  8  -f-  cos  <f  cos  d  —  2  cos  <p  cos  d  sin2  £  t  =  sin  h 
But 

sin  <p  sin  d  -\-  cos  <f>  cos  d  =  cos  (<p  —  <5)  or  cos  (<5  —  <p) 

Hence,  if  we  put 

Ci  =  V  —  d,          or  d  =  a  —  ? 
the  above  equation  may  be  written 

cos  d  =  sin  h  -f  cos  <f>  cos  d  (2  sin2  £  f)  (282) 

If  the  star  does  not  change  its  declination,  £L  is  the  zenith 
distance  of  the  star  at  its  meridian  passage  ;  and,  being  found  by 


234  LATITUDE. 

this  equation,  we  then  have  the  latitude  as  from  a  meridian 
observation  by  the  formula 

V  =  d  +  :15     or     <p  =  d  -  Ct 

according  as  the  zenith  is  north  or  south  of  the  star. 

When  the  star  changes  its  declination,  this  method  still  holds 
if  we  take  d  for  the  time  of  observation,  as  is  evident  from  our 
formulae,  in  which  d  is  the  declination  at  the  instant  when  the 
true  altitude  is  h. 

To  compute  the  second  member,  a  previous  knowledge  of  the 
latitude  is  necessary.  As  the  term  cos  <p  cos  d  (2  sin2  J  t)  de- 
creases with  /,  if  the  observations  are  not  too  far  from  the 
meridian,  the  error  produced  by  using  an  approximate  value  of 
<p  will  be  relatively  small,  so  that  the  latitude  found  will  be  a 
closer  approximation  than  the  assumed  one ;  and  if  the  computa- 
tion be  repeated  with  the  new  value,  a  still  closer  approximation 
may  be  made,  and  so  on  until  the  exact  value  is  found. 

This  method  is  only  convenient  where  the  computer  is  pro- 
vided with  a  table  of  natural  sines  and  cosines,  as  well  as  a  table 
of  log.  versed  sines,  or  the  logarithmic  values  of  2  sin2  J  t. 

EXAMPLE.— Same  as  Example  1,  Art.  164.  h  =  57°  35'  11", 
d  =  -f  2°  51'  30",  t  =  lh  20wt  30*.  Approximate  value  of  <p=  -23°. 

log  (2  sin2  ,}  t)  8.785726 
log  cos  <p          9.964026 

nat.  sin  h  0.844201  log  cos  d  9.999459 

nat.  no.       0.056132 log          ^749211 

nat.  cos  :t  0.900333 

C,  =  —  25°  47'  54"  (zenith  south  of  sun.) 

9  =  +    2   51  30 

<p  =  —  22~56~24 

differing  but  13"  from  the  true  value,  although  the  assumed 
latitude  was  in  error  nearly  4'.  Repeating  the  computation  with 
-  22°  56'  24"  as  the  approximate  latitude,  we  find  <p=  -  22°  56'  11", 
exactly  as  in.  Art.  164. 

169.  (B.)  We  may  also  compute  directly  the  reduction  of  the 
observed  altitude  to  the  meridian  altitude.  Putting 

Jil  —  meridian  altitude  =  90°  —  C4 


CIRCUMMERIDIAX    ALTITUDES.  235 

the  formula  (282)  gives 

sin  Aj  —  sin  h  =  2  cos  <f>  cos  d  sin2  £  t 
But  we  have 

sin  hi  —  sin  h  =  2  cos  \  (hl  -f-  A)  sin  *  (ht  —  A) 

and  hence 

.  cos  <p  cos  5  sin2  -J  t 

sin  i  (A,  —  h)  =  -      ———  ——  (283) 

cos  \  (hL  -f  A) 

which  gives  the  difference  A1  —  A,  or  the  correction  of  h  to  reduce 
it  to  A!  ;  but  it  requires  in  the  second  member  an  approximate 
value  both  of  <p  and  of  hl9  the  latter  being  obtained  from  the 
assumed  value  of  <p  by  the  equation  h{  =  90°  -  (<p  —  o);  or,  if 
the  zenith  is  south  of  the  star,  by  the  equation  A1  —  90°  —  (d  —  <p). 

EXAMPLE. — Same  as  the  above. 

d   =         2°  51'  30"  log  sin2  *  t               8.484696 

Approx.          <p   =  —  23   00  00  log  cos  <p                9.964026 

Cj  =       25   51  30  log  cos  d                 9.999459 

"                  \  =       64     8  30  log  sec  i  (A,  -f  A)  0.312573 

«     i  (^  _|_  /t)  =       60   51  50  log  sin  J  (Ax  —  A)  8J60754 

/ti_A   =         6   36  33 

A   =       57    35  11  8  =         2°  51'  30" 

A,  =       64   11  44  d  =  —  25   48  16 
^  i=  —  22   56  46 

This  method  does  not  approximate  so  rapidly  as  the  preceding, 
but  the  objection  is  of  little  weight  when  the  observations  are 
very  near  the  meridian.  On  the  other  hand,  it  has  the  great 
advantage  of  not  requiring  the  use  of  the  table  of  natural  sines. 

170.  (C.)  Circummeridian  altitudes. — When  a  number  of  altitudes 
are  observed  very  near  the  meridian,*  they  are  called  circum- 
meridian  altitudes.  Each  altitude  reduced  to  the  meridian  gives 
nearly  as  accurate  a  result  as  if  the  observation  were  taken  on  the 
meridian. 

An  approximate  method  of  reducing  such  observations  with 
the  greatest  ease  is  found  by  regarding  the  small  arc  £  (AL  —  A) 
as  sensibly  equal  to  its  sine ;  that  is,  by  putting 

sin  I  (At  —  A)  =  i  (A,  —  A)  sin  1" 
*  How  near  to  the  meridian  will  be  determined  in  Art.  175. 


236  LATITUDE. 

and  taking  AL  for  i  (/^  -f  A),  from  which  it  differs  very  little,  so 
that  (283)  may  be  put  under  the  form 


cos  7tj  sin  1" 

The  value  in  seconds  of 

2  sin2  £  £ 


(284) 


m  = 


sn 


is  given  in  Table  V.  with  the  argument  t.  If  7t',  A",  A"',  &c.  are 
the  observed  altitudes  (corrected  for  refraction,  etc.);  <',  <",  <"', 
&c.,  the  hour  angles  deduced  from  the  observed  clock  times  ; 
m1,  m",  mf",  £c.,  the  values  of  m  from  the  table  ;  and  we  put  the 
constant  factor 


(285) 


COS  <f>  COS  8 COS  <p  COS  d 

cos  Aj  sin  Cx 

we  have  A5  =.  A'    4*  Am' 

A!  ==  A"    4- 

A,  =  A'"  4- 
&c. 

and  the  mean  of  all  these  equations  gives 

h'  4-  h"  4-  h'"  +  etc.  m'  4-  m"  4-  m'"  4-  &c. 

A    —  . _ _j_  A — 

n  n 

in  which  n  is  the  number  of  observations ;  or 

\  =  hQ  4-  Am0  (286) 

in  which  A0  denotes  the  mean  of  the  observed  altitudes  corrected 
for  refraction,  £c.,  and  mQ  the  mean  of  the  values  of  m. 

When  A1  has  been  thus  found,  the  latitude  is  deduced  as  from 
any  meridian  altitude,  only  observing  that  for  the  sun  the  de- 
clination to  be  used  is  that  which  corresponds  to  the  mean  of 
the  times  of  observation,  as  has  already  been  remarked  in  Art. 
168. 

EXAMPLE. — At  the  U.  S.  Naval  Academy,  1849  June  22,  cir- 
cummeridian  altitudes  of  ft  Ursae  Minoris  were  observed  with  a 
Troughton  sextant  from  an  artificial  horizon,  as  in  the  following 
table.  The  times  were  noted  by  a  sidereal  chronometer  which 


CIRCUMMERIDIAN   ALTITUDES. 


237 


was   fast  lm  45'. 7.     The   index   correction   of  the   sextant  was 
-  14'  58",  Barometer,  30.81  inches,  Att.  Therm.  65°  F.,  Ext. 
Therm.  64°  F. 

The  right  ascension  of  the  star  was   14*  51W  14'.0 
Chronometer  fast  -f-  I    45  .7 

Chronometer  time  of  star's  transit      14   52    59  .7 

The  hour  angles  in  the  column  t  are  found  by  taking  the  differ- 
ence between  each  observed  chronometer  time  and  this  chro- 
nometer time  of  transit. 


2  Ait,  # 

108°  3<y  40" 

39  50 

40  40 

41  0 
41  0 
40  30 
40  20 
40  0 
40  0 
39  20 

Mean  108  40  14 

Ind.  corr.   -  14  58 

108  25  16 


Refr. 


54  12  38 
—  42  .0 
-f  21  .5 


^  =  _  35  47  42  .5 
6  =  74  46  36  .9 
6  =  38  58  54  .4 


Chronom. 
14A  45™  47'. 

47  1. 

48  54.5 
51  29.5 
54  36.5 

56  22. 

57  43. 

58  47.5 
0  17.5 
2  10. 


15 


t 

lm  12'.  7 
5  58.7 
4  5.2 
1  30.2 
1  36.8 

3  22.3 

4  43.3 

5  47.8 
7  17.8 
9  10.3 


Assumed  0  =  38°  59'  0" 

6  =  74  46  36  .9 


Approx.  Ci=  35  47  36  .9 


log  cos  0  9.8906 

log  cos  d  9.4193 

logcosecd  0.2329 

log  A  9.5428 

log  m0  1.7898 

log  Am9  1.3326 


REMARK  1.  —  The  reduction  Ax  —  h  increases  as  the  denominator 
of  A  decreases,  that  is,  as  the  meridian  zenith  distance  decreases. 
The  preceding  method,  therefore,  as  it  supposes  the  reduction  to 
be  small,  should  not  be  employed  when  the  star  passes  very  near 
the  zenith,  unless  at  the  same  time  the  observations  are  restricted 
to  very  small  hour  angles.  It  can  be  shown,  however,  from  the 
more  complete  formulae  to  be  given  presently,  that  so  long  as 
the  zenith  distance  is  not  less  than  10°,  the  reduction  computed 
by  this  method  may  amount  to  4'  30"  without  being  in  error 
more  than  V  ;  and  this  degree  of  accuracy  suffices  for  even  the 
best  observations  made  with  the  sextant. 


238  LATITUDE. 

REMARK  2.~If  in  (284)  we  put  sin %t  =  l£  sin  1".  t  (t  being  in 
seconds  of  time),  we  have 

=_  cos  y  cos  8    2M  g.n        ^2  ^  ^ 
cos  Aj  2 

in  which  a  denotes  the  product  of  all  the  constant  factors.  It 
follows  from  this  formula  that  near  the  meridian  the  altitude  varies 
as  the  square  of  the  hour  angle,  and  not  simply  in  proportion  to  the 
time.  Hence  it  is  that  near  the  meridian  we  cannot  reduce  a 
number  of  altitudes  by  taking  their  mean  to  correspond  to  the 
mean  of  the  times,  as  is  done  (in  most  cases  without  sensible 
error)  when  the  observations  are  remote  from  the  meridian. 
The  method  of  reduction  above  exemplified  amounts  to  sepa- 
rately reducing  each  altitude  and  then  taking  the  mean  of  all 
the  results. 

171.  (D.)  Circummeridian  altitudes  more  accurately  reduced. — The 
small  correction  which  the  preceding  method  requires  will  be 
obtained  by  developing  into  series  the  rigorous  equation  (282). 
This  equation,  when  we  put  £  —  90°  —  h  =  true  zenith  distance 
deduced  from  the  observation,  may  be  put  under  the  form 

cos  C  =  cos  Cx  —  2  cos  <p  cos  d  sin2  £  t 

which  developed  in  series*  gives,  neglecting  sixth  and  higher 
powers  of  sin  \  t. 


*  If  we  put  y  =  2  cos  0  cos  6  sin2  £  t,  the  equation  to  be  developed  is 

cos  {  =  cos  £1  —  y  (a) 

in  which  £t  is  constant  and  £  may  be  regarded  as  a  function  of  y  ;  so  that  by  MAC- 
LAURIN'S  Theorem 


in  which  (/),  I  —  1,  &c.  denote  the  values  of  fy  and  its  differential  coefficients  when 

\dy  I 

y  =  0.     The  equation  (a)  gives,  by  differentiation, 


dr          1 


dy  dy        sin 

cos  t    dC  cot  C 


&c. 


dy*  sin2  £    dy 


CIRCUMMERIDIAN    ALTITUDES,  239 


cos  v  cos  8  2  sin2  %  t  ,  /  cos  ^  cos  S\*  2  cot  d  sin* 


—  - 


sin2  %  t    ,   /  cos  ^  cos  S\* 
-4-  i  -  I 
in  1"          \      sin  d       / 


sin  sin  1"  sin  d  sin  I" 


(£00) 


By  this  formula,  first  given  by  DELAMBRE,  the  reduction  to 
the  meridian  consists  of  two  terms,  the  first  of  which  is  the  same 
as  that  employed  in  the  preceding  method,  and  the  second  is  the 
small  correction  which  that  method  requires.  These  two  terms 
will  be  designated  as  the  "  1st  Reduction"  and  "  2d  Reduction." 
Putting 

2  sin2  if  2  sin*  Jf 

m=—  n  =  -  - 

sin  1"  sin  1" 

cos  (p  cos  «5 
A  =—  B  =  A*  cot  d 

sin  d 
we  have 

d  =  C  —  Am  -f  En  (289) 

If  a  number  of  observations  are  taken,  we  have  a  number  of 
equations  of  this  form,  the  mean  of  which  will  be 


in  which  £0  is  the  arithmetical  mean  of  the  observed  zenith  dis- 
tances, m0  and  n0  the  arithmetical  means  of  the  values  of  m  and 
n  corresponding  to  the  values  of  t.  The  values  of  n  are  also 
given  in  Table  Y. 

Having  found  £r,  wre  have  the  latitude,  as  before,  by  the  formula 


in  which  we  must  give  ^  the  negative  sign  when  the  zenith  is 
south  of  the  star,  and  it  must  be  remembered  that  for  the  sun 
(or  any  object  whose  proper  motion  is  sensible)  3  must  be  the 
mean  of  the  declinations  belonging  to  the  several  observations, 


But  when  y  —  0  we  have,  by  (a),  £  =  ^,  so  that  (6)  becomes 


sm  £j       2  sin2  £t  sin3  £a 

Restoring  the  value  of  y,  this  gives  the  development  used  in  the  text,  observing  that 
as  £  and  ^  are  supposed  to  be  in  seconds  of  arc,  the  terms  of  the  series  are  divided 
by  sin  1"  to  reduce  them  to  the  same  unit. 


240  LATITUDE. 

or,  which  is  the  same,  the  declination  corresponding  to  the  mean 
of  £be  times  of  observation.* 

Finally,  if  the  star  is  near  the  meridian  below  the  pole,  the 
hoar  angles  should  be  reckoned  from  the  instant  of  the  lower 
transit.  Recurring  to  the  formula 

cos  C  =  sin  <p  sin  d  -\-  cos  <p  cos  8  cos  t 

in  which  t  is  the  hour  angle  reckoned  from  the  upper  transit, 
we  observe  that  if  this  angle  is  reckoned  from  the  lower  transit 
we  must  put  180°  —  t  instead  of  £,  or  —  cos  t  for  -f-  cos  t,  and  then 

we  have 

cos  C  =  sin  <p  sin  <5  —  cos  <p  cos  <5  cos  t 

and,  substituting  as  before, 

cos  t  =  1  —  2  sin2  $  t 
this  gives 

cos  C  =  —  cos  (<p  -f-  fl)  -f  2  cos  <f>  cos  d  sin2  £  t 

or,  since  for  lower  culminations  we  have  £\  =  180°  —  (<p  -f-  #) 
and  cos-fj  =  —  cos  (<p  +  <J), 

cos  C  =  cos  C,  -f  2  cos  <f>  cos  J  sin2  J  f 
which  developed  gives 

cos  c?  cos  #    2  sin2  \  t       /  cos  p  cos  (5  f   2  cot  rt  sin4  $  f 


•sin2  £  f       /  cos  9?  cos  5  \2 
jin  1"         \      sin  Ci      / 


sin  d  sin  1"         \      sin  Ci      /  sin  1" 

or 

^  =  c  -f  Am  +  Bn  (sub  polo)  (290) 

which  is  computed  by  the  same  table,  but  both  first  and  second 
reductions  here  have  the  same  sign. 

If  a  star  is  observed  with  a  sidereal  chronometer  the  daily 
rate  of  which  is  so  small  as  to  be  insensible  during  the  time  of 

*  To  show  that  the  mean  declination  is  to  be  used,  we  may  observe  that  for  each 
observation  we  have  put  ^  =  0  —  tf,  and  that  if  6',  d",  &c.,  are  the  several  declina- 
tions, the  several  equations  of  the  form  (289)  will  give 

0  =  6'  +  C  —  Am1  -f  A1  cot  f,  n' 
0  =  6"  -f-  C"  —  Am'1  -j-  A*  cot  (,  n" 

&c., 
the  mean  of  which,  if  6  =  mean  of  ff',  d",  &c.,  will  be 


CIRCUMMERIDIAN    ALTITUDES.  241 

the  observations,  the  hour  angles  t  are  found  by  merely  taking 
the  difference  between  each,  noted  time  and  the  chronometer 
time  of  the  star's  transit,  as  in  the  example  of  Article  170.  But 
if  we  wish  to  take  account  of  the  rate  of  the  chronometer,  it  can 
be  done  without  separately  correcting  each  hour  angle,  as  fol- 
lows: Let  STbe  the  rate  of  the  chronometer  in  24/l  (3T  being 
positive  for  losing  rate,  Art.  137) ;  then,  if  t  is  the  hour  angle 
given  directly  by  the  chronometer,  and  tf  the  true  hour  angle, 
we  have 

f  :  t  =  24*  :  24»  —  dT  =  86400* :  86400'  —  dT 
whence 


Instead  of  sin  \t  we  must  use  sin  %t' ';  for  which  we  shall  have, 
with  all  requisite  precision, 

sin  K  = 
Hence,  if  we  put 

~~86400j 

we  shall  have 

,    cos  <p  cos  8  2  sin2  J  t 

Am  =  k i 

sin  d          sin  1" 

so  that  if  we  compute  A  by  the  formula 

cos  <p  cos  8 
sin  d 

we  can  take  m  =  — —      -  for  the  actual  chronometer  intervals, 
sm  1" 

and  no  further  attention  to  the  rate  is  required. 

The  factor  k  can  be  given  in  a  small  table  with  the  argument 
"rate,"  in  connection  with  the  table  for  m,  as  in  our  Table  Y. 

If  a  star  is  observed  with  a  mean  time  chronometer,  the  inter- 
vals are  not  only  to  be  corrected  for  rate,  but  also  to  be  reduced 

VOL.  I.— 16 


242  LATITUDE. 

from  mean  to  sidereal  intervals  by  multiplying  them  by  p.  = 
1.00273791  (Art.  49) ;  so  that  for  sin2  J  t  we  must  substitute  k  sin2 
(2  •  /^)>  or>  with  sufficient  precision,  kfj?  sin2  J  t. 

If  the  sun  is  observed  with  a  mean  time  chronometer,  the  in- 
tervals are  both  to  be  corrected  for  rate  and  reduced  from  mean 
solar  to  apparent  solar  intervals.  The  mean  interval  differs 
from  the  apparent  only  by  the  change  in  the  equation  of  time 
during  the  interval,  and  this  change  may  be  combined  with  the 
rate  of  the  chronometer.  Denoting  by  dE  the  increase  of  the 
equation  of  time  in  24*  (remembering  that  E  is  to  be  regarded 
as  positive  when  it  is  additive  to  apparent  time),  and  by  dTthe 
rate  of  the  chronometer  on  mean  time,  we  may  regard  d  T —  dE 
as  the  rate  of  the  chronometer  on  apparent  time.  Instead  of 
the  factor  k  we  shall  then  have  a  factor  /:',  which  is  to  be  found 
by  the  formula 


.,  r     i     r 
i  __*"•-'* 

86400     J 


which  may  be  taken  from  the  table  for  k  by  taking  d  T —  dE  as 
the  argument. 

Finally,  if  the  sun  is  observed  with  a  sidereal  chronometer, 

we  must  multiply  sin2  J  t  not  only  by  kf  but  by  — 2 . 

Denoting  fj?  by  i  and  -3  by  £',  these  rules  may  be  collected,  for 
the  convenience  of  reference,  as  follows : 


.     COS  <p  COS  d 

Star  by  sidereal  chron.,       A  =  k  •  - 

sin  Ct 

Star  by  mean  time  chron.,  A  =  ki.GOSVc°9(J  [logt  =  0.002375] 

sin  d 

(29r 

cos  <p  cos  3 
Sun  by  mean  time  chron.,  A  —K  •  — 

sin  C± 

Sun  by  sidereal  chron.,       4  =  K  i'  C°8  y  C°S  *[logf  =  9.997625] 

sin  *" 

for  which  log  &  will  be  taken  from  Table  Y.  with  the  argument 
rate  of  the  chronometer  =  d T\  and  log  kf  from  the  same  table 


CIRCUMMERIDIAN    ALTITUDES.  243 

with  the  argument  dT  --  dE '=  daily  rate  of  the  chronometer 
diminished  by  the  daily  increase  of  the  equation  of  time. 

EXAMPLE. — 1856  March  15,  at  a  place  assumed  to  be  in  lati- 
tude 37°  49'  K  and  longitude  122°  24'  W.,  suppose  the  fol- 
lowing zenith  distances  of  the  sun's  lower  limb  to  have  been 
observed  with  an  Ertel  universal  instrument,*  Barom.  29.85 
inches,  Att.  Therm.  65°  F.,  Ext.  Therm.  63°  F.  The  chrono- 
meter, regulated  to  the  local  mean  time,  was,  at  noon,  slow 
llm  20*.8,  with  a  daily  losing  rate  of  6*.6. 

Obs'd  zen.  dist.  Chronometer.  t  m                      n 

40°    8'40".7  23*  37**  35'.  -19-58-.8  783".3  1".49 

40      216.5  42      3.  -15  30.8  472.4  0.54 

39    5728.3  46    29.5  —11  4.3  240.6  0.14 

39    54  17  .2  50    46.5  --6  47.3  90  .5  0  .02 

39    52  33  .  55    16 .  -  2  17 .8  10  .4  0  .00 

39    52  34  .5  0     0    37 .5  +   3  3 .7  18  .4  0  .00 

39    54  28  .6  5    13 .  7  39  .2  115  .0  0  .03 

39  58     9  .8  9    49.5  12  15.7  295.1  0.21 

40  3     0  .3  14      8 .          16    34 .2  538  .9  0  .70 
40      936.             18    31.          20    57.2           861.4  1.80 


Means  39    59  18  .5  f0=  -f   0    29.1  m0  =  342  .60  w0=0  .49 

The  equation  of  time  at  the  local  noon  being  -f-  8"*  54*.6,  we 
have 

Mean  time  of  app.  noon  =   0*  8TO  54*.6 
Chronometer  slow  =       11    20 .8 

Chr.  time  of  app.  noon  =  23  57    33  .8 

The  difference  between  this  and  the  observed  chronometer 
times  gives  the  hour  angles  t  as  above. 

The  mean  of  the  hour  angles  being  -f  29s.  1,  the  declination  is 
to  be  taken  for  the  local  apparent  time  07t  Owl  29*.  1,  or  for  the 
Greenwich  mean  time  March  15,  Sh  18"*  59*.7;  whence 

3  =—    1°  48'  8".8 
(Approximate)  <p  =  -f-  37    49  0  . 
"  C,=       39    37  8  .8 

The  increase  of  the  equation  of  time  in  24*  is  dE  =  —  17'.4, 

*  See  Vol.  II.,  Altitude  and  Azimuth  Instrument,  for  the  method  of  observing  the 
zenith  distances. 


244  LATITUDE. 

and,  the  chronometer  rate  being  3T  =  -f-  6*.6,  we  have  3T —  dE 
=  -f-  24*.0,  with  which  as  the  argument  urate"  in  Table  Y.  we 
find  log  k'  ==  0.00024. 

The  computation  of  the  latitude  is  now  carried  out  as  follows  : 

log  cos  <f>       9.89761  Mean  observed  zen.  dist.  Q  =  39°  59'  18". 5 

log  cos  3       9.99979  r~p=  +  41.8 

log  cosec  Ct  0.19540  log  A1          0.1861                S=  16     6  .5 

log  kf            0.00024  log  cot  C,     0.0821           Am^  =  —  7     4  .4 

log  A            0.09304  log  B            0.2682             Bn0  =  +  0  .9 

log  m0           2.53479  log  n0           9.6902                C,  =  39  36  50  73 

logAmQ        2L62783  log  Bn<>        9.9584                 d=  —I  48     8  .8 

?=  37  48  41  .5 

The  assumed  value  of  <p  being  in  error,  the  value  of  A  is  not 
quite  correct;  but  a  repetition  of  the  computation  with  the  value 
of  <p  just  found  does  not  in  this  case  change  the  result  so  much 
as  0".l. 

172.  (E.)  Gauss's  method  of  reducing  circummeridian  altitudes  of 
the  sun. — The  preceding  method  of  reduction  is  both  brief  and 
accurate,  and  the  latitude  found  is  the  mean  of  all  the  values 
that  would  be  found  by  reducing  each  observation  separately. 
This  separate  reduction,  however,  is  often  preferred,  notwith- 
standing the  increased  labor,  as  it  enables  us  to  compare  the 
observations  with  each  other,  and  to  discuss  the  probable  error 
of  the  final  result;  and  it  is  also  a  check  against  any  gross  error. 
But,  if  we  separately  reduce  the  observations  by  the  preceding 
method,  we  must  not  only  interpolate  the  refraction  for  each 
altitude,  but  also  the  declination  for  each  hour  angle.  GAUSS 
proposed  a  method  by  which  the  latter  of  these  interpolations  is 
avoided.  He  showed  that  if  we  reckon  the  hour  angles,  not 
from  apparent  noon,  but  from  the  instant  when  the  sun  reaches  its 
maximum  altitude,  we  can  compute  the  reduction  by  the  method 
above  given,  and  use  the  meridian  declination  for  all  the  observa- 
tions. This  method  is,  indeed,  not  quite  so  exact  as  the  preced- 
ing; but  I  shall  show  how  it  may  be  rendered  quite  perfect  in 
practice  by  the  introduction  of  a  small  correction. 

In  the  rigorous  formula 

cos  C  =  sin  <z>  sin  d  -f-  cos  <p  cos  3  cos  t 


CIRCUMMERIDIAN    ALTITUDES.  245 

8  is  the  declination  corresponding  to  the  hour  angle  t.     If  then 

A<5  =  the  hourly  increase  of  the  declination,  positive  when 

the  sun  is  moving  northward, 
dl  =  the  declination  at  noon, 

and  if  t  is  expressed  in  seconds  of  time,  we  have 


where,  since  A£  never  exceeds  60",  x  will  not  exceed  30"  so  long 
as  t  <  30'".     Hence  we  may  substitute,  with  great  accuracy, 

sin  £  =  sin  S1  -(-  cos  dl  sin  x 
cos  3  =  cos  (Sj  —  sin  dl  sin  x 

and  the  above  formula  becomes 


C  =  sin  <p  sin  ^  -f-  cos  <p  cos  dl  cos  t  -f-  sin  (9?  —  d^)  sin  x 
-f-  2  cos  <f  sin  dl  sin2  £  £  sin  x 


The  last  term  is  extremely  small,  rarely  affecting  the  value  of  £ 
by  as  much  as  0".l;  and  since  x  is  proportional  to  the  hour 
angle,  and  therefore  has  opposite  signs  for  observations  on  differ- 
ent sides  of  the  meridian,  the  effect  of  this  term  will  nearly  or 
quite  disappear  from  the  mean  of  a  series  of  observations  pro- 
perly distributed  before  and  after  the  meridian  passage.  Now, 
we  have 


sin  x  = =  15  t  sin  1"  • 


3600  54000 

Let 


54000    cos  <p  cos  <5t 
then,  taking 

15  t  sin  1"  =  sin  t  -f  J  sin3  t 
we  have 

f   .  .    ,  ,,      .  COS  <f>  COS  S 

sm  #  =  (sm  t  -\-  i  sin3  f)  sin  #  •  - 

sin(?>-<g 

and  the  formula  for  cos  £  becomes,  by  omitting  the  last  term, 

cos  C  =  sin  <p  sin  ^  -j-  cos  <p  cos  ^(cos  t  -f-  sin  f  sin  #) 
-j-  g  cos  ^  cos  (\  sin'f  sin  # 


246  LATITUDE. 

The  last  term  involving  sin3  t  multiplied  by  the  small  quantity 
sia  #is  even  less  than  the  term  above  rejected.  Like  that,  also, 
it  has  opposite  signs  for  observations  on  different  sides  of  the 
meridian,  and  will  not  affect  the  mean  result  of  a  properly 
arranged  series  of  observations.  Rejecting  it,  therefore,  our  for- 
mula becomes 

cos  C  =  sin  <p  sin  d±  -f-  cos  <f>  cos  dl  cos  (t  —  #) 
-j-  2  cos  (p  cos  d1  sin2  \  # 

The  last  term  here  must  also  be  rejected  if  we  wish  to  obtain  the 
method  as  proposed  by  GAUSS  ;  but,  as  it  is  always  a  positive 
term  and  affects  all  the  observations  alike,  I  shall  retain  it,  espe- 
cially as  it  can  be  taken  into  account  in  an  extremely  simple 
manner. 

The  maximum  value  of  cos  f,  which  corresponds  to  the 
maximum  altitude,  is  given  immediately  by  the  above  formula 
by  putting  t  =  &.  Hence  $  is  the  hour  angle  of  the  maximum  altitude. 
Putting 

f  =  t  —  & 
we  have 

cos  C  =  cos  (</>  —  <Sj)  —  2  cos  <p  cos  dl  sin2  J  f 
-j-  2  cos  y  cos  dl  sin2  £  $ 
Let 

,  cos  <p  cos  (\      2  sin2  \  # 

1  +  sin  fa  _  y  '     sin  1" 

then  our  formula  becomes 

cos  C  —  cos  (<p  —  <$')  —  2  cos  <p  cos  dl  sin2  J  tf 

This  equation  is  of  the  same  form  as  that  from  which  (288)  was 
obtained,  and  therefore  when  developed  gives 


__  r       cos  y  cos  \   2  sin2  \t'       /  cos  <p  cos  d1  V  2  cot  C,  sin*  J  f 
sin  Cj  sin  1"         \       sin  C,       /  sin  I" 

in  which  £x  —  ^  —  or.     Putting  then,  as  before, 

^cos^cos^         5^2  (292) 

sin  Ct 

• 

and  taking  m  and  r?  from    Table  Y.,  or  their  logarithms  from 
Table  VI.,  with  the  argument  t'9  which  is  the  hour  angle  reckoned 


CIRCUMMERIDIAN    ALTITUDES.  247 

from  the  instant  the  sun  reaches  its  maximum  altitude,  we  have 

Ct  =  C  —  Am  -f  En  (293) 

Since  £t  differs  from  the  latitude  by  the  constant  quantity  8',  its 
value  found  from  each  observation  should  be  the  same.  Taking 
its  mean  value,  we  have 

The  angle  $,  being  very  small,  may  be  found  with  the  utmost 
precision  by  the  formula 

—  =  [9.40594]  —  (294) 


810000  sin  I"    A  A 

which  gives  &  in  seconds  of  the  chronometer  when  A  has  been 
computed  by  the  formula  (292). 

The  most  simple  method  of  finding  the  corrected  hour  angles 
tf  will  be  to  add  $  to  the  chronometer  time  of  apparent  noon, 
and  then  take  the  difference  between  this  corrected  time  and 
each  observed  time. 

If  we  put  d'  =  dj_  -f-  y,  we  have 

^.isinlif 
sin  V 

which  requires  only  one  new  logarithm  to  be  taken;  namely,  the 
value  of  log  m  from  Table  VI.  with  the  argument  $.  We  then 
have,  finally, 

9  =  Ci  +  \  +  y  (296) 

EXAMPLE. — The  same  as  that  of  the  preceding  article.  We 
have  there  employed  the  assumed  latitude  37°  49' ;  but,  since  even 
a  rough  computation  of  two  or  three  observations  will  give  a 
nearer  value,  let  us  suppose  we  have  found  as  a  first  approxima- 
tion tp  =  37°  48'  45".  With  this  and  the  meridian  declination 
d^=  -  1°  48'  9".2,  and  log  k'  =  0.00024  as  before,  we  now  find, 
by  (292), 

log  A  =  0.09310  log  S  =  0.2683 

We  have  also  there  found  the  chronometer  time  of  apparent 


248  LATITUDE. 

noon  =  23*  57m  33*.8.  We  now  take  from  the  Ephemeris  ±3  = 
+  59".22,  and  hence,  by  (294), 

log  A*  1.7725 

ar.  co.  log  A  9.9069 

const,  log  9.4059 

*  =  +  12'.2  log  #  1.0853 

Hence  the  chronometer  time  of  the  maximum  altitude  i» 
23*  57m  33*.8  -f  12'.2  =  23*  57m  46s,  which  gives  the  hour  angles 
/'  as  below : 


f 

log  m 

log  Am 

log  n 

log  Bn 

-  20-  11*. 

2.90274 

2.99584 

0.1900 

0.4583 

15  43. 

2.68558 

2.77868 

9.7557 

0.0240 

11  16.5 

2.39718 

2.49028 

9.1776 

9.4459 

6  59.5 

1.98216 

2.07526 

8.3487 

8.6170 

-  2  30. 

1.08891 

1.18201 

-f-  2  51.5 

1.20525 

1.29835 

7  27. 

2.03730 

2.13040 

8.4553 

8.7236 

12  3.5 

2.45551 

2.54861 

9-2955 

9.5638 

16  22. 

2.72077 

2.81387 

9.8260 

0.0943 

20  45. 

2.92677 

3.01987 

0.2381 

0.5064 

The  refraction  may  be  computed  from  the  tables  first  for  a  mean 
zenith  distance,  and  then  with  its  variation  in  one  minute  (which 
will  be  found  with  sufficient  accuracy  from  the  table  of  mean 
refraction)  its  value  for  each  zenith  distance  is  readily  found. 
The  parallax,  which  is  here  sensibly  the  same  (—  5". 54)  for  all 
the  observations,  is  subtracted  from  the  refraction,  and  the  results 
are  given  in  the  column  r  —  p  of  the  following  computation. 
The  numbers  in  the  3d  and  4th  columns  are  found  from  their 
logarithms  above ;  and  the  last  column  contains  the  several 
values  of  the  minimum  zenith  distance  of  the  siin's  lower  limb, 
formed  by  adding  together  the  numbers  of  the  preceding  columns. 
To  the  mean  of  these  we  then  apply  the  sun's  semidiameter,  the 
meridian  declination,  and  the  correction  yy  which  are  all  constant 
for  the  whole  series  of  observations. 


CIRCUMMERIDIAN   ALTITUDES.  249 


Obs' 

d  c 

r  — 

P 

Am 

Bn 

Ci 

40° 

8'  40" 

.7 

+  42".l 

—  16 

'  30". 

5       -f- 

2".  9 

39°  52'  55" 

2 

40 

2 

16 

.5 

41 

.9 

10 

0  . 

7 

1  .1 

58 

.8 

39 

57 

28 

.3 

41 

.8 

5 

9  . 

2 

0  .3 

61 

.2 

39 

54 

17 

2 

41 

.7 

1 

58  . 

9 

0  .0 

60 

.0 

39 

52 

33 

41 

.6 

0 

15  . 

2 

0  .0 

59 

.4 

39 

52 

34 

.5 

41 

.6 

0 

19  . 

9 

0  .0 

56 

.2 

39 

54 

28 

.6 

41 

.7 

2 

15  . 

0 

0  .1 

55 

.4 

39 

58 

9 

.8 

41 

.8 

5 

53  . 

7 

0  .4 

58 

.3 

40 

3 

0 

.3 

41 

.9 

10 

51  . 

4 

1  .2 

52 

.0 

40 

9 

36 

42 

.1 

17 

26  . 

8 

3  .2 

54 

.5 

(Lower  limb) 

Mean 

Ci  = 

39    52  57 

.10 

\f\CT 

2  sin2  J  # 

8.9090 

Semidiumeter  = 

16     6 

.49 

JOg 

sin  1" 

81  = 

-  1    48     9 

.20 

log 

A 

0.0931 

y  = 

-h              0 

.10 

log 

y 

9.0021 

<p  = 

37    48  41 

.51 

This  result  agrees  precisely  with  that  found  before.  If  we  suppose 
all  the  observations  to  be  of  the  same  weight,  we  can  now  deter- 
mine the  probable  error  of  observation.  Denoting  the  difference 
between  each  value  of  f  t  and  the  mean  of  all  by  y,  and  the  sum 
of  the  squares  of  v  by  [t-y],  according  to  the  notation  used  in  the 
method  of  least  squares,  we  have 

V  VV 

-  1".9  3.61 

-{-  1  .7  2.89  Mean  error  of  a  single  observa- 

4.1  16.81  I  [«,] 

-  = 


+  2  .9  8.41 

+  2  .3  5.29  Mean  error  of  the  final  value  of 

-  0  .9  .81  2  89 

-  1  .7  2.89  9  =  -7^  =  0".91 

-f  1  .2  1.44 

-  5  .1  26.01 
kr.  2  .6  6.76 
n  =  10,  [vv]  =  74.92 

Probable  error  of  a  single  obs.  ==  2".89  X  0.6745  =  1".95 
«  "      of  ?  =0  .91  X  0.6745  =  0  .61 

It  must  not  be  forgotten  that  the  probable  error  1".95  here 
represents  the  probable  error  of  observation  only  :  a  constant  error 
of  the  instrument,  affecting  all  the  observations,  will  form  no 
part  of  this  error;  and  the  same  is  true  of  an  error  in  the 
refraction. 


250  LATITUDE. 

173.  For  the   most   refined   determinations   of  the   latitude, 
standard  stars  are  to  be  preferred  to  the  sun.    Their  declinations 
are  somewhat  more  precisely  known  ;  the  instrument  is  in  night 
observations  less  liable  to  the  errors  produced   by  changes  of 
temperature    during   the    observations  ;    constant    instrumental 
errors    and   errors  of  refraction  may  be  eliminated  to  a  great 
extent  by  combining  north  and  south  stars  ;  or  errors  of  declina- 
tion may  be  avoided  by  employing  only  circumpolar  stars  at  or 
near  their  upper  and  lower  culminations.    In  general,  errors  of 
circummeridian  altitudes  are  eliminated  under  the  same  condi- 
tions as  those  of  meridian  observations,  since  the  former  are 
reduced  to  the  meridian  with  the  greatest  precision.     See  the 
next  following  article. 

For  a  great  number  of  nice  determinations  of  the  latitude  by 
circummeridian  altitudes  of  stars  north  and  south  of  the  zenith 
and  of  circumpolar  stars,  see  PUISSANT,  Nouvelle  Description  Geo- 
metrique  de  la  France. 

174.  Effect  of  errors  of  zenith  distance,  declination,  and  time,  upon 
the  latitude  found  by  circummeridian  altitudes.  —  Differentiating  (289), 
regarding  A  as  constant,  and  neglecting  the  variations  of  the 
last  term,  which  is  too  small  to  be  sensibly  affected  by  small 
errors  of  t,  we  have,  since  dtp  =  d^  -f-  dS, 


(15  dt) 


sin  1" 

The  errors  d?  and  dd  affect  the  resulting  latitude  by  their  whole 
amount.  The  coefficient  of  dt  has  opposite  signs  for  east  and 
west  hour  angles;  and  therefore,  if  the  observations  are  so  taken 
as  to  consist  of  a  number  of  pairs  of  equal  zenith  distances  east 
and  west  of  the  meridian,  a  small  constant  era>r  in  the  hour 
angles,  arising  from  an  imperfect  clock  correction,  will  be  elimi- 
nated in  the  mean.  This  condition  is  in  practice  nearly  satisfied 
when  the  same  number  of  observations  are  taken  on  each  side 
of  the  meridian,  the  intervals  of  time  between  the  successive 
observations  being  made  as  nearly  equal  as  practicable. 

An  error  in  the  assumed  latitude  which  affects  A  is  eliminated 
by  repeating  the  computation  with  the  latitude  found  by  the  first 
computation.  An  error  in  the  declination  which  would  affect  A 
is  not  to  be  supposed. 


LIMITS    OF    CIRCUMMERIDIAN    ALTITUDES.  251 

175.  To  determine  the  limits  within  which  the  preceding  methods  oj 
reducing  circummeridian  altitudes  are  applicable. — First.  In  the 
method  of  Art.  170  we  employ  only  the  "  first  reduction"  (=  Am), 
which  is  the  first  term  of  the  more  complete  reduction  expressed 
by  (288).  The  error  of  neglecting  the  "  second  reduction"  (=  Bn) 
increases  with  the  hour  angle,  and  if  this  method  is  to  be  used  it 
becomes  necessary  to  determine  the  value  of  the  hour  angle  at 
which  this  reduction  would  be  sensible.  We  have 

Bn  —  A2  cot  C,  2  81"4  *  *- 
sin  1" 

whence  if  we  put  b  for  Bn  and 

F  =  I/  i  sin  1"  tan  C,    % 


we  derive 


sin2  \t  =  —  i/b  (298) 


Since  £t  =  <p  —  8,  F  and  A  are  but  functions  of  <p  and  3  ;  and 
therefore  by  this  formula  we  can  compute  the  values  of  t  for 
any  assigned  value  of  6,  and  for  a  series  of  values  of  <p  and  d. 
Table  VILA  gives  the  values  of  t  in  minutes  computed  by  (298) 
when  b  =  V.  That  is,  calling  ^  the  tabular  hour  angle  and  t 
the  hour  angle  for  any  assigned  limit  of  error  6,  we  have 


sin2  ^tl  =  —  sin2  i  t  =  sin2  J  f  y'b 

A 

As  the  limits  are  not  required  with  great  precision,  we  may  sub- 
stitute for  the  last  equation  the  following  : 

t  =  tlt/b 

If  we  take  b  =  0".l,  we  have  jJ/6  =  0.56,  or  nearly  J  :  hence  the 
limiting  hour  angle  at  which  the  second  reduction  amounts  to  0".l  is 
about  J  the  angle  given  in  Table  VILA. 

EXAMPLE.  —  How  far  from  the  meridian  may  the  observations 
in  the  example  p.  237  be  extended  before  the  error  of  the 
method  of  reduction  there  employed  amounts  to  1"?  With 
p=:  +  39°,  3=  +  75°,  Table  VILA  gives  ^  =  30'».  Hence 


252 


LATITUDE. 


the  method  is  in  that  example  correct  within  1"  if  the  observa- 
tions are  taken  within  30W  of  the  meridian,  and  correct  within 

O'M  if  they  are  taken  within  15m  of  the  meridian. 

« 

Second. — In  the  more  exact  methods  of  reduction  given  in 
Arts.  171  and  172,  we  have  neglected  the  last  term  of  the 
development  given  in  the  note  on  page  239,  which  may  be  called 
a  "  third  reduction."  Denoting  it  by  <?,  we  have 

4/1H-3  cot2  :  \ 
c  =  —    -  — l  I  A3  sin6  £  t 

3  \        sin  1"        / 

whence,  if  we  put 


,/    fBinl" 

"  \i  +  3cot2 ct 

we  deduce 

sin2  Jf  =  —  l/c  (299) 

A 

+ 

Table  YII.B  gives  the  values  of  /,  computed  by  this  formula,  for 
c  =  V.  Denoting  the  tabular  value  of  t  by  tl9  we  have 

sin2  J  f4  =  —  sin2  *  t  =  sin2  *  t±  l/c 

A 

or,  with  sufficient  accuracy  in  most  cases, 

t  =  tli/c 

For  c  =  O'M  we  have  \/c  =  0.68,  or  nearly  f ;  and  hence  the 
limiting  hour  angle  at  which  the  third  reduction  (omitted  in  our 
most  exact  methods)  would  amount  to  O'M  is  about  f  the  angle 
given  in  Table  VII.B. 

EXAMPLE. — How  far  from  the  meridian  may  the  observations 
in  the  example  p.  243  be  extended  before  the  error  of  the 
method  of  reduction  there  employed  amounts  to  O'M  ?  With 
<p  =  38°,  S  =  —  2°,  Table  VII.B  gives  t,  ==  39*1,  and  f  of  this 
is  t  =  26™:  so  that  the  method  is  in  that  example  correct  within 
1"  when  the  observations  are  taken  within  39W  of  the  meridian ; 
and  it  is  correct  within  O'M  when  the  observations  are  taken 
within  26W  of  the  meridian. 

The  limiting  hour  angle  for  a  given  limit  of  error  diminishes 


BY   THE    POLE    STAR.  253 

rapidly  with  the  zenith  distance ;  and  hence  in  general  very  small 
zenith  distances  are  to  be  avoided.  But  the  observations  may  be 
extended  somewhat  beyond  the  limits  of  our  tables,  since  the 
errors  which  affect  only  the  extreme  observations  are  reduced  in 
taking  the  mean  of  a  series. 

FOURTH    METHOD. — BY  THE    POLE    STAR. 

176.  The  latitude  may  be  deduced  with  accuracy  from  an  alti- 
tude of  the  pole  star  observed  at  any  time  whatever,  when  this 
time  is  known.  The  computation  may  be  performed  by  (281); 
but  when  a  number  of  successive  observations  are  to  be  reduced, 
the  following  methods  are  to  be  preferred.  If  we  put 

p  =  the  star's  polar  distance, 
we  have,  by  (14), 

sin  h  =  sin  <p  QOS  p  -f-  cos  <p  sin  p  cos  t 

in  which  the  hour  angle  t  and  the  altitude  h  are  derived  from 
observation  and  <p  is  the  required  latitude.  Now,  p  being  small 
(at  present  less  than  1°  30'),  we  may  develop  (f  in  a  series  of 
ascending  powers  of  p,  and  then  employ  as  many  terms  as  we 
need  to  attain  any  given  degree  of  precision.  The  altitude 
cannot  differ  from  the  latitude  by  more  than  p:  if,  then,  we  put 

y>  =  h  —  x 

x  will  be  a  small  correction  of  the  same  order  of  magnitude  as  p. 
We  have* 

sin  <p  —  sin  (h  —  x)  =  sin  h  —  x  cos  h  —  4  x1  sin  h  -j-  i  x3  cos  h  -f-  &c. 
cos  <p  =  cos  (h  —  x)  =  cos  h  -j-  x  sin  h  —  i  a:2  cos  h  —  J  x3  sin  h  -j-  &c. 
tinp  =p—  £;?*+  &c. 
cos  p  =  1  —  -J-  p*  -j-  &c. 

which  substituted  in  the  above  formula  for  sin  h  give 

sin  h  =  sin  h  —  x  cos  h  -j-  p  cos  t  cos  h  —  J-  (.r2  —  2  .rp  cos  t  -\-p*}  sin  h  -f  &c. 

and  from  this  we  obtain  the  following  general  expression  of  the 
correction : 

*  PI.  Trig.  (403)  and  (406). 


254  LATITUDE. 

x  =  p  cos  t  —  J  (.T2  —  2  ;r/>  coa  £  -j-  />2)  tan  A 
-f  i  (*3  —  3  x2  p  cos  £  +  3  x^?2  —  ^3  cos  0 
-f  2!4  (#*  —  4  a;3  p  cos  t  -f-  6  x2  j92  —  4  .rp3  cos  f  +  ^4)  tan  A 
—  &c.  («) 

For  a  first  approximation,  we  take 

x  =  p  cos  t  (6) 

and,  substituting  this  in  the  second  term  of  (#),  we  find  for  a 
second  approximation,  neglecting  the  third  powers  of  p  and  x, 

x  =  p  cos  t  —  IP*  sin2 1  tan  h  (c) 

Substituting  this  value  in  the  second  and  third  terms  of  (a),  we 
find,  as  a  third  approximation, 

x  =  p  cos  t  —  J  p*  sin2 1  tan  h  -f-  -J  />3  cos  £  sin2  f  (Y7) 

This  value,  substituted  in  the  second,  third,  and  fourth  terms  of 
(a\  gives,  as  a  fourth  approximation, 

x  =  p  cos  t  —  ^  p2  sin2 1  tan  ft  -f-  ^  p*  cos  £  sin2  f  —  |  p4  sin4  #  tan3  h 

_|_  ^i_  p*  (4  _  9  sin2  f)  sin2 1  tan  /t  (6) 

In  these  formulae,  to  obtain  x  in  seconds  when  p  is  given  in 
seconds,  we  must  multiply  the  terms  in  p2,  p3,  and  p4  by  sin  1", 
sin2  1",  sin3  1",  respectively. 

In  order  to  determine  the  relative  accuracy  of  these  formulae, 
let  us  examine  the  several  terms  of  the  last,  which  embraces  all 
the  others.  The  value  of  /,  which  makes  the  last  term  of  (e)  a 
maximum,  will  be  found  by  putting  the  differential  coefficient 
of  (4  —  9  sin2 1)  sin2 1  equal  to  zero ;  whence 

4  sin  t  cos  t  (2  —  9  sin2  f)  =  0 

which  is  satisfied  by  t  =  0,  t  —  90°,  or  sin2 1  =  f,  the  last  of  which 
alone  makes  the  second  differential  coefficient  negative.  The 
maximum  value  of  the  term  is,  then,  J?  p4  sin3  V  tan  h,  which 
for  p  =  1°  30r  =  5400"  is  0".0018  tan  h.  This  can  amount  to 
0".01  only  when  h  is  nearly  80°, — that  is,  when  the  latitude  is 
nearly  80°.  This  term,  therefore,  is  wholly  inappreciable  in 
every  practical  case. 


BY   THE    POLE    STAR.  255 

The  term  \p4  sin3  V  sin4  £tan3  h  is  a  maximum  for  sin  t  =  1, 
in  which  case,  for  p  ===  5400",  it  is  0".0121  tan3  h.  This,  amounts 
to  0".l  when  A  ==  64°,  and  to  V.  when  h  =jj  77°. 

For  the  maximum  of  the  term  J  />3  sin2 1"  cos  £  sin2 1  we  have, 
hy  putting  the  differential  coefficient  of  cos  t  sin2 1  equal  to  zero, 

sin  £  (2  —  3  sin2  f)  =  0 

which  gives  sin2  £  =  f ,  and  consequently  cos  t  =  j/J- ;  and  hence 
the  maximum  value  of  this  term  is  f  ju3  sin2  1"  i/J  =  0".475. 
Since  the  maximum  values  of  this  and  the  following  terms  do 
not  occur  for  the  same  value  of  /,  their  aggregate  will  evidently 
never  amount  to  V  in  any  practical  case. 

Hence,  to  find  the  latitude  by  the  pole  star  within  1",  we  have  the 

formula 

</>  =  h  —  p  cos  t  -f  ±pz  sin  1"  sin2 1  tan  h  (300) 

The  hour  angle  t  is  to  be  deduced  from  the  sidereal  time  0 
of  the  observation  and  the  star's  right  ascension  a,  by  the 
formula 

t  =  0  — tt 

To  facilitate  the  computation  of  the  formula  (300),  tables  are 
given  in  every  volume  of  the  British  Nautical  Almanac  and  the 
Berlin  Jahrbuch;  but  the  formula  is  so  simple  that  a  direct 
computation  consumes  very  little  more  time  than  the  use  of 
these  tables,  and  it  is  certainly  more  accurate. 

EXAMPLE. — (From  the  Nautical  Almanac  for  1861). — On  March 
6,  1681,  in  Longitude  37°  W.,  at  7/l  43"1  35*  mean  time,  suppose 
the  altitude  of  Polaris,  when  corrected  for  the  error  of  the  in- 
strument, refraction,  and  dip  of  the  horizon,  to  be  46°  17'  28" : 
required  the  latitude. 

Mean  time  7*  43m  35*. 

Sid.  time  mean  noon,  March  6,  22   56    47  .9 

Reduction  for  7*  43W  35'  1    16  .2 

Reduction  for  Long.  2*  28"*  24.3 

Sidereal  time  0  =    6  42      3.4 

March  6,  p  =  1°  25'  32".7  a  =    1     7    39.0 

t    =    5  34    24.4 
=  83°  36'  6" 


256  LATITUDE. 

Hence,  by  formula  (300), 


3.71035  logp2  7.4207 

log  cos  t         9.04704  log  sin2  1  9.9946 

log  1st  term  2.75739  log  tan  h  0.0196 

log  £sinl"  4.3845 

h  =  46°  17'  28"  log  2d  term  1.8194 

1st  term  =  -       9  32  .0 
2d     "      =  -f      1     6  .0 

?r=46      9     2  .0 
By  the  Tables  in  the  Almanac,  ?  =  46°  9'  V 

177.  If  we  take  all  the  terms  of  (e)  except  the  last,  which  we 
have  shown  to  be  insignificant,  we  have  the  formula 

<f>  =  h  —  p  cos  t  -j-  iF*  sin  1"  sin2  1  tan  h 

-  I  p*  sin2  1"  cos  t  sin2  1  -f  |^4  sin3  1"  sin*  t  tan3  h        (301) 

which  is  exact  within  0".01  for  all  latitudes  less  than  75°,  and 
serves  for  the  reduction  of  the  most  refined  observations  with 
first-class  instruments. 

If  we  have  taken  a  number  of  altitudes  in  succession,  the 
separate  reduction  of  each  by  this  formula  will  be  very  laborious. 
To  facilitate  the  operation,  PETERSEN  has  computed  very  con- 
venient tables,  which  are  given  in  SCHUMACHER'S  Hiilfstafelh, 
edited  by  WARNSTORFF.  These  tables  give  the  values  of  the 
following  quantities  : 

a  =  p0  cos  t  -j-  ^  p03  sin2  1"  cos  t  sin2  1 

13  =  ±p*  sin  1"  sin2  1 

X  =  lp(p2  —  j»02)  sin2  1"  cos  t  sin2  1 

/JL  =  I  p4  sin3  1"  sin*  t  tan3  h 

* 

in  which  p0  =  1°  30'  =  5400".     Then,  putting 


Po 

log  A  =  logp  —  3.7323938 
the  formula  (301)  becomes 

p  =  h  —  (A*  +  X)  +  A*0  tan  h  -f 


BY   TWO    ALTITUDES.  257 

Putting  then 

x  =  Ao,  -f  I 
y  =  A*fi  tan  h  -j-  y. 
we  have 


(302) 


or,  when  the  zenith  distance  f  is  observed, 


#  =  ^4a  -f  A  ^ 

y  t=  4s/9  cot  C  +  ja  >  (303) 

90°  -  ?  =  C  +  *  -  y  J 

The  first  table  gives  a  with  the  argument  /;  the  second,  ft  with 
the  argument  t;  the  third,  ^  with  the  arguments  p  and  t;  and 
the  fourth,  /*  with  the  arguments  y  and  ^,  ^  being  used  for  h  \\\ 
so  small  a  term. 

To  reduce  a  series  of  altitudes  or  zenith  distances  by  these 
tables,  we  take  for  h  or  f  the  mean  of  the  true  altitudes  or 
zenith  distances  ;  for  a  and  /9  the  means  of  the  tabular  numbers 
corresponding  to  the  several  hour  angles,  with  which  we  find 
Aa  and  J.2/3  cot  £.  The  mean  values  of  the  very  small  quanti- 
ties A  and  /jt  are  sensibly  the  same  as  the  values  corresponding  to 
the  mean  of  the  hour  angles  ;  so  that  /  is  taken  out  but  once  for 
the  arguments  polar  distance  and  mean  hour  angle,  and  //  is 
taken  with  the  arguments  <p  and  the  approximate  value  of  y  = 
A2fl  cot  f.  Illustrative  examples  are  given  in  connection  with 
the  tables. 

FIFTH  METHOD.  —  BY  TWO  ALTITUDES  OF  THE  SAME  STAR,  OR  DIF. 
FERENT  STARS,  AND  THE  ELAPSED  TIME  BETWEEN  THE  OBSERVA- 
TIONS. 

178.  Let  S  and  S'9  Fig.  25,  be  any  two  points  of  Fig.  25. 
the  celestial  sphere,  Z  the  zenith  of  the  observer, 
Pthe  pole.  Suppose  that  the  altitudes  of  stars  seen 
at  S  and  $',  either  at  the  same  time  or  diiferent 
times,  are  observed,  and  that  the  observer  has  the 
means  of  determining  the  angle  SPS'  ;  also  that 
the  right  ascensions  and  declinations  of  the  stars 
are  known.  From  these  data  we  can  find  both  the  latitude  and  the 
local  time.  A  graphic  solution  (upon  an  artificial  globe)  is  indeed 
quite  simple,  and  it  will  throw  light  upon  the  analytic  solution. 
"With  the  known  polar  distances  of  the  stars  and  the  angle  SPS', 

VOL.  I.—  17 


258  LATITUDE. 

let  the  triangle  SPS'  be  constructed.  From  S  and  S'  as  poles 
describe  small  circles  whose  radii  (on  the  surface  of  the  sphere) 
are  the  given  zenith  distances  of  S  and  S'  :  these  small  circles  inter- 
sect in  the  zenith  Z  of  the  observer,  and,  consequently,  determine 
the  distance  PZ,  or  the  co-latitude,  and  at  the  same  time  the  hour 
angles  ZPS  and  ZPS',  from  either  of  which  with  the  star's  right 
ascension  we  deduce  the  local  time.  This  graphic  solution  shows 
clearly  that  the  problem  has,  in  general,  two  solutions  ;  for  the 
small  circles  described  from  S  and  S'  as  poles  intersect  in  two 
points,  and  thus  determine  the  zenith  of  another  observer  who 
at  the  same  instants  of  time  might  have  observed  the  same  alti- 
tudes of  the  same  stars.  The  analytic  solution  will,  therefore, 
also  give  two  values  of  the  latitude  ;  but  in  practice  the  ob- 
server always  has  an  approximate  knowledge  of  the  latitude, 
which  generally  suffices  to  distinguish  the  true  value. 

Let  us  consider  lirst  the  most  general  case. 

(A.)   Two  different  stars  observed  at  different  times.  —  Let 

h,  hf  =  the  true  altitudes,  corrected  for  refraction,  &o., 
T,  T'  =  the  clock  times  of  observation, 
A  T,  &Tr  =  the  corresponding  corrections  of  the  clock, 
a,  a'  —  the  right  ascensions,  and 
8,  d'  =  the  declinations  of  the  stars  at  the  times  of  the 

observations  respectively, 
#,  f  —  the  hour  angles  of  the  stars  at  the  times  of  the 

observations  respectively, 
X  =  f  —  t  =  the  difference  of  the  hour  angles, 
<p  =  the  latitude  of  the  observer  : 

then  we  have,  if  the  clock  is  sidereal, 


t  =  T  -f  A      —  a 

f  =  T  +  A  T  —  a' 

A  =  (T  —  T)  +  (AT'—  AT)  —  (a'—  a)  (304) 

a  formula  which  does  not  require  a  knowledge  of  the  absolute 
values  of  &T  and  AT7',  but  only  of  their  difference;  that  is,  of 
the  rate  of  the  clock  in  the  interval  between  the  two  obser- 
vations. 

If  the  clock  is  regulated  to  mean  time,  the  interval  T'  •  -  T  + 
A  T'  —  A  T  is  to  be  converted  into  a  sidereal  interval  by  adding 
the  acceleration,  Art.  49. 

We  have  next  to  obtain  formula  for  determining  f  and  t  or  t' 


BY   TWO    ALTITUDES. 


259 


from  the  data  A,  A',  £,  ^',  and  L     I  shall  give  two  general  solu- 
tions, the  first  of  which  is  the  one  commonly  known. 

First  Solution.— Let  the  observed  points  S  and  S'  be  joined 
by  an  arc  of  a  great  circle  SS'.  In  the  triangle  PSS'  there  are 
given  the  sides  PS=  90°  —  S,  PS'  =  90°  —  d'9  and  the  angle  SPS' 
=  ^  from  which  we  find  the  third  side  SSf  =  J5,  and  the  angle 
PS'S=  P,  by  the  formulae  [E  of  Art.  10] 

cos  B  =  sin  df  sin  8  -{-  cos  <S'  cos  8  cos  A 
sin  jB  cos  P  =  cos  5'  sin  8  —  sin  3'  cos  8  cos  /I 
ein  B  sin  P  =  cos  8  sin  A 

or,  adapted  for  logarithmic  computation, 


m  sin  M=  sin  5 

m  cos  M  =  cos  <5  cos  A 

cos  B  =  m  cos  (Jf  - 

sin  .B  cos  P  =  m  sin  (Jf  - 

sin  B  sin  P  =  cos  8  sin  A 


(305) 


In  the  triangle  ZSS'  there  are  now  known  the  three  sides 
ZS  =  90°  —  A,  ZS'  =  90°  —  hf,  SS'  =  B,  and  hence  the  angle 
ZSfS  =  Q,  by  the  formula  employed  in  Art.  22: 


cos  A'  sin 


(806) 


Now,  putting 


there  are  known  in  the  triangle  PZS'  the  sides  PS'  =  90°  —  S', 
ZS'  =  $Q°  —  h',  and  the  angle  PS'Z=q,  from  which  the  side 
PZ=  90°  —  y>,  and  the  angle  S'PZ=  t',  are  found  by  the  formulae 

sin  <f>  =  sin  8f  sin  A'  -f  cos  «5'  cos  A'  cos  q 
cos  ?>  cos  tf  =  cos  5'  sin  h'  —  sin  8'  cos  hf  cos  <? 
cos  9?  sin  f  =  cos  A'  sin  q 

or,  adapted  for  logarithmic  computation, 


n  sin  ^V  =  sin  hr 

n  cos  N=  cos  A'  cos  g' 

sin  <p  =  n  «os  (JV —  ^' 
cos  ^>  cos  f  =  n  sin  (JV —  #' 
cos  sin  t'  =  cos  h'  sin 


.(307) 


260  LATITUDE. 

The  formulae  (305)  and  (307)  leave  no  doubt  as  to  the  quadrant 
in  which  the  unknown  quantities  are  to  be  taken.  But  we  may 
take  the  radical  in  (306)  with  either  the  positive  or  the  negative 
sign,  and  J  §,  therefore,  either  in  the  first  or  fourth  quadrant. 
If  we  take  J  Q  always  in  the  first  quadrant,  the  values  of  q  will  be 

q  =  P+  Q 

and  either  value  may  be  used  in  (307) ;  whence  two  values  of  <p 
and  tf.  That  value  of  <p,  however,  which  agrees  best  with  the 
known  approximate  latitude  is  to  be  taken  as  the  true  value. 
There  is  also  another  method  of  distinguishing  which  value  of  q 
will  give  the  true  solution ;  for,  if  A'  and  A  are  the  azimuths  of 
8'  and  S9  we  have  in  the  triangle  ZSS'  the  angle  SZS'  =  A'  —  A, 
and 

sin  0  =  sin  (A'  — A)  - 

sin  B 

in  which  cos  h  and  sin  B  are  always  positive.  Hence  Q  is  posi- 
tive or  negative  according  as  A'  —  A  is  less  or  greater  than  180°. 
The  observer  will  generally  be  able  to  decide  this :  the  only  cases 
of  doubt  will  be  those  where  A'  and  A  are  very  nearly  equal  or 
where  A'  —  A  is  very  nearly  180°  ;  but,  as  we  shall  see  hereafter, 
these  cases  are  very  unfavorable  for  the  determination  of  the 
latitude,  and  are,  therefore,  always  to  be  avoided  in  practice 

If  the  great  circle  88'  passes  north  of  the  zenith,  we  shall  have 
A'  —  A  negative,  or  greater  than  180° :  hence  we  have  also  this 
criterion  :  we  must  take  q  =  P  —  Q  or  q  =  P  -f  Q  according  as  the 
great  circle  SSf  passes  south  or  north  of  the  zenith. 

Second  Solution.— Bisect  the  arc  SS',  Fig.  25,  in  T\  join  PT 
and  ZT,  and  put 

C  =  ST  =  S'T, 

D  =  the  declination  of  T=  90°  —  P77, 

H  =  the  altitude  of  T  =  90°  —  ZT, 

T  =  the  hour  angle  of  T  =  ZPT, 
P  =  the  angle  PTS, 
Q  =  the  angle  ZTS, 

q  =  the  angle  PTZ. 

We  have  in  the  triangle  PSS'  [Sph.  Trig.  (25)] 

Bin1  C  =  sin2 %  (3  —  <5')  cos2  M  +  cos2  £  (3  -f  <5')  sin2  J  ; 


BY   TWO    ALTITUDES.  261 

or,    adapted   for   logarithmic   computation,   by   introducing   an 
auxiliary  angle  E, 

sin  G  sin  E  =  sin  %  (3  —  d')  cos  H  }       _ 

sin  C  cos  E  ==  cos  *  (<5  +  d')  sin  U  j 

In  the  triangle  SPT  we  have  the  angle  PTS=P,  and  there- 
fore in  the  triangle  S'PTwQ  have  the  angle  PTS'  =  180°  —  P, 
the  cosine  of  which  will  be  =  —  cos  P:  hence,  from  these 
triangles  we  have  the  equations 


sin  D  cos  C  -j-  cos  D  sin  C  cos  P  =  sin 
sin  .Z>  cos  (7  —  cos  D  sin  (7  cos  P  —  sin 


whence 


2  sin  D  cos  C  =  sin  d  -f  sin  d' 
2  cos  D  sin  C  cos  P  =  sin  d  —  sin  d' 

sin  £  (<5  4-  <$")  cos  £  (<5  —  5') 

sin  D  =  -  — - - 

cos  C 

(309) 

cos  P  =  CQS  *  (*  +  O  8in  *  (*  —  ^ 
cos  J9  sin  (7 

which  determine  D  and  P  after  (7  has  been  found  from  (308). 

In  precisely  the  same  manner  we  derive  from  the  triangles 
ZTS  and  ZTS'  the  equations 


T_       sin  £  (h  -4-  h'~)  cos  $  (h  —  A') 

sin  .ff  = — — 

cos  C 

cos  J  (A  -f-  A')  sin  £  (A  —  ^') 


C08      = 


cos  If  sin  (7 


Then  in  the  triangle  PTZwe  have  the  angle  PTZ,  by  the 
formula 


and  hence  the  equations 

sin  <p  =  sin  D  sin  If  -f-  cos  D  cos  H  cos 
cos  ^  cos  r  =  cos  D  sin  H  —  sin  D  cos  H  cos 
cos  <p  sin  T  =  cos  H  sin 


262  LATITUDE. 

To  adapt  tliese  for  logarithmic  computation,  let  /9  and  j  be  deter- 
mined by  the  conditions* 

cos  ft  sin  Y  =  cos  H  cos  q  \ 

cos  ft  cos  y  =  sin  H  I    (311) 

sin  ft  =  cos  H  sin  q  J 

then  y>  and  r  are  found  by  the  equations 

sin  <p  =  cos  ft  sin  (JD  -j-  f)  ~\ 

cos  <p  cos  r  =  cos  ft  cos  (D  -f-  r)  V    (312) 

cos  ^  sin  T  =  sin  ft  J 

To  find  the  hour  angles  t  and  t',  let 

tf  =  T  —  i  (£'  -f-  0  f 

then,  since  J  A  =  J  (^ —  £),  we  have 

J  A  -f  x  =  r  —  t  =  the  angle  TP£, 

1     3    y»  -4?    —  4-  V\  p    fill  (1*1  A     ^/7  Z^,^' 

and  from  the  triangles  PTtf  and  PT£'  we  have 

sin  (i  ^  -f-  x} sin  P  sin  ( J  J  —  ar) sin  P 

sin  (7  cos  d  sin  (7  cos  £' 

whence 

sin  (J  A -for)  —  sin(JA  —  x)       cos<S'  —  cos  d 


sin  (H  -f  x)  +  sin  (J  x  —  xi)       cos  <J'  + 
and,  consequently, 

tan  x  =  tan  *  (d  -f  <5')  tan  |  (d  —  <$')  tan  *  Jl  (313) 

Hence,  finally, 


As  in  the  first  solution,  the  value  of  q  will  become  =  P  +  § 
when  the  arc  joining  the  observed  places  of  the  stars  passes  north 
of  the  zenith. 

EXAMPLE.  —  1856  March  5,  in  the  assumed  Latitude  89°  17'  N. 
and  Longitude  5*  6m  36*  "W.,  suppose  the  following  altitudes 

*  The  equations  (811)  can  always  be  satisfied,  since  the  sum  of  their  squares  gives 
the  identical  equation  1=1. 


BY   TWO    ALTITUDES.  263 

(already  corrected  for  refraction)  to  have  been  obtained ;  the 
time  being  noted  by  a  mean  solar  chronometer  whose  daily  rate 
was  10*. 32  losing.  The  star  Arc-turns  was  not  far  from  the  prime 
vertical  east  of  the  meridian ;  Spica  was  near  the  meridian. 

Arcturus,  h  =       18°    6'  30"  Chronometer  T  =    9*  40M  24'.8 

Spica,        h'  =       40      4  43  «  T'  =  U  38    16.7 

T'^  T  ;=    4   57    51 .9 

d  ==  +  19°  55'  44".6  Corr,  for  rate       =         +2.1 

d'  s=  —  10    24  39  .5  Bed.  to  sid.  time  =__     -f-   48  .9 

Sid.  interval         =    4   58   42 .9 

a   =       14*    9"  6-.70  a  —  a'  =    0   51    29.1 

a'  =       13    17  37 .72  /I  =    5   50   12 .0 

=  87°  33'    0". 

According  to  our  first  solution,  we  obtain, 

by  (305),  B  =  91°  15'  52".5         P  =  69°  57'  54".7 

and,  by  (306),  Q  =  64    51  24  .8 

whence  #  =    5      6  29  .9 

Then,  by  (307),  we  find 

<p  =  39°  17'  20"        t'  =  5°  3'  0"  =   0*  20"  12'. 

a'  =  13  17   37  .72 


Sidereal  time  of  the  observation  of  Spica  =  13  37    49  .72 
Sidereal  time  at  mean  Greenwich  noon     =  22  53   39  .83 


14  44     9.89 

Acceleration  for  14*  44*  9'.89  =  —     2   24  .85 

«  longitude       ==  —         50  .23 

Mean  time  of  the  observation  of  Spica      =  14  40   54  .81 

Chronometer  correction  at  that  time,  A  7"  =  -j-     2m  38'.11 

According  to  the  second  solution,  we  prepare  the  quantities 


J  /I  =  43°  46'  30"     J(fl+d')=   4°45'32".6     J(A  +  A')=      29°    5'36".5 
i(<5  —  i')  =  15   1012.1     £(A—  A')  =  —  10    59    6.5 

with  which  we  find,  by  (308),  (309),  and  (310), 

log  tan  E  =  9.437854  D  =         6°  34'  32".0 

log  sin  C  =  9.854225  P  =       68    27  22  .2 

log  cos  C  =  9.844639  Q  =     108    35  12  .1 

log  sin  ff=  9.834176  q  =  —  40      7  49  .9 
log  cos  ff=  9.863785 


264  LATITUDE. 

(The  auxiliaries  C and  H are  not  themselves  required:  we  take 
their  cosines  from  the  table,  employing  the  sines  as  arguments.) 
We  now  find,  by  (311),  (312),  (313),  and  (314), 

log  sin  £  =  W9.673029  r  =  322°  30'  51" .3 

log  cos  /?  =    9.945532  x  =      1    14  21  .3 

r  =  39°  18'    4".0  T—x=  321    16  30 

D  _j_  r  =  45    52  36  .0  =    21*  25™    6' 

<p  =  39    17  20  .  J  A  =      2    55      6 

*  =    18    30  0 

f=     .0    20  12 

agreeing  precisely  with  the  results  of  the  first  solution. 

179.  In  the   observation  of  lunar  distances,  as  we  shall  see 
hereafter,  the  altitudes  of  the  moon  and  a  star  are  observed  at 
the   same  instant  with  the   distance  of  the  objects.     The  ob- 
served distance  is  reduced  to  the  true  geocentric  distance,  which 
is  the  arc  B  o£  the  above  first  solution,  or  2  (7  of  the  second.     The 
observation  of  a  lunar  distance  with  the  altitudes  of  the  objects 
furnishes,  therefore,  the  data  for  finding  the  latitude,  the  local 
time,  and  the  longitude. 

180.  (B.)  A  fixed  star  observed  at  two  different  limes. — In  this  case 
the  declination  is  the  same  at  both  observations,  and  the  general 
formulae  of  the  preceding  articles  take  much  more  simple  forms. 
The  hour  angle  ^  is  in  this  case  merely  the  elapsed  sidereal  time 
between    the    observations,    the    formula   (304),    since   <x  =  a', 
becoming  here 

1  =  (Tf  —  T}  +  (AT7'— AT)  (315) 

Putting  8f  for  3  in  (308)  and  (309),  we  find  E"—  0,  cos  P=  0, 
P  —  90° ;  and  C  and  D  are  found  by  the  equations 

sin  C  =  cos  d  sin  £  I,       sin  D  = (316) 

cos  C 

Since  we  have  q  =  P —  Q  =  90°  —  §,' the  last  two  equations  of 
.(311)  give 

sin  0  =  cos  H  cos  <2,        cos  Y  =  sin  Jf  sec  /9 


BY   TWO    ALTITUDES.  265 

which,  by  (310),  become* 


.        _  cos  &  (h  -f  A')  sin  $  (h  —  hr) 
sin  C 


sin  £  (h  -4-  A')  cos  £  (h  —  h'* 
cos  r  — 


(317) 


cos  /?  cos  C 

Then  <p  and  r  are  found  by  (312),  or  rather  by  the  following : 
sin  <p  =  cos  ft  sin  (D  -j-  p) 


(318) 


tan  /?  sin  /3 

tan  T  =  -       —     or  sin  r  =  — 

cos(Z>-fr)  cos? 

The  hour  angles  at  the  two  observations  are 


Here  f,  being  determined  by  its  cosine,  may  be  either  a  posi- 
tive or  a  negative  angle,  and  we  obtain  two  values  of  the  latitude 
by  taking  either  D  -f  f  or  D  —  f  in  (318).  The  first  value  wrill 
be  taken  when  the  great  circle  joining  the  two  positions  of  the 
star  passes  north  of  the  zenith ;  the  second,  when  it  passes  south 
of  the  zenith. 

The  solution  may  be  slightly  varied  by  finding  D  by  the 
formula 

tan  D  =    tan  *  (320) 

COS  z  A 

obtained  directly  from  the  triangle  P TS  (Fig.  25),  which  is  right- 
angled  at  T  when  the  declinations  are  equal.  We  can  then  dis- 
pense with  C  by  writing  the  formulae  (317)  as  follows : 

cos  J  (fc  +  h')  sin  \  (h  —  h') 


Bill    p  :  — 

sin  £  ( 

cos  d  sin  £  A 

h  -f-  h')  cos  £  (h  —  A')  sin  D 

(321) 

cos  ft  sin  d 

*  The   formulae  (315),   (316),  and  (317)  are  essentially  the  same  as    those  first 
given  for  this  case  by  M.  CAILLET  in  his  Manuel  du  Navigateur,  Nantes,  1818. 


266  LATITUDE. 

181.  (C.)  TJie  sun  observed  at  two  different  times. — In  this  ease 
the  hour  angle  X  is  the  elapsed  apparent  solar  time.  If  then  the 
times  77and  Tf  are  observed  by  a  mean  solar  chronometer,  and 
the  equation  of  time  at  the  two  observations  is  e  and  e'  (positive 
when  additive  to  apparent  time),  we  have 

1  =  (Tf—  T)  +  (AT'—  AT)  —  (e'—e)  (322) 

Taking  then  the  declinations  o  and  3'  for  the  two  times  of  obser- 
vation, we  can  proceed  by  the  general  methods  of  Art.  178. 

But,  as  the  declinations  differ  very  little,  we  can  assume  their 
mean — i.e.  the  declination  at  the  middle  instant  between  the 
observations — as  a  constant  declination,  and  compute  at  least  an 
approximate  value  of  the  latitude  by  the  simple  formulae  for  a 
fixed  star  in  the  preceding  article.  We  can,  however,  readily 
correct  the  resulting  latitude  for  the  error  of  this  assumption. 
To  obtain  the  correction,  we  recur  to  the  rigorous  formulae  of  our 
second  solution  in  Art.  178.  The  change  of  the  sun's  declination 
being  never  greater  than  V  per  hour,  we  may  put  cos  J  (d  —  df) 
=  1.  Making  this  substitution  in  (308)  and  (309),  and  putting  d 
for  \  (d  -f  d')  so  that  d  will  signify  the  mean  of  the  declinations, 
and  A#  for  \  (d'  —  8)  so  that  A$  will  signify  one-half  the  increase 
of  the  sun's  declination  from  the  first  to  the  second  observation 
(positive  when  the  sun  is  moving  northward),  we  shall  have 


sin  A'5 
tan  E  =  — 


cos  d  tan  J  A 

But  A£  diminishes  with  A,  so  that  J?  always  remains  a  small 
quantity  of  the  same  order  as  A£  ;  and  hence  we  may  also  put 
cos  E=\.  Thus  the  second  equation  of  (308)  gives 

sin  C  =  cos  <5  sin  I  A 
and  the  first  of  (309) 

.        __  sin  d 

cos  G 

which  are  the  same  as  (316),  as  given  for  the  case  where  the 
declination' is  absolutely  invariable.  Hence  no  sensible  error  is 
produced  in  the  values  of  C  and  D  by  the  use  of  the  mean  de- 


BY    TWO    ALTITUDES.  267 

clination.     But  by  the  second  equation  of  (309)  P  will  no  longer 
be  exactly  90°  :  if  then  we  put 

P  =  90°  +  AP 
we  have,  by  that  equation, 

cos  d  sin  A'S  sin  A<5 


sin  AP  = 


cos  D  sin  C       cos  D  sin  £  JL 
or  simply 


cos  D  sin  £  A 
Then,  since  q  =  P  —  §,  we  have 

<?  =  90°  —  §  +  AP 
The  rigorous  formula  for  the  latitude  is 

sin  <p  =  sin  D  sin  //  -f-  cos  D  cos  H  cos  q 

in  which  when  we  use  the  mean  declination  we  take  q  =  90°  — 
Q:  therefore,  to  find  the  correction  of  <p  corresponding  to  a  cor- 
rection of  q  =  AP,  we  have  by  differentiating  this  equation,  D 
and  H  being  invariable, 

cos  <p  .  A^>  =  —  cos  D  cos  H  sin  q  .  AP 

A<5  cos  H  cos  Q 
sin  \  A 

We  have  found  in  the  preceding  article  sin  $  =  cos  H  cos  Q; 
and  hence 


cos     sin  t 


(323) 


In  the  case  of  the  sun,  therefore,  we  compute  the  approximate 
latitude  tp  by  the  formulae  (816),  (317),  and  (318),  employing  for  d 
the  mean  declination.  We  then  find  A^  by  (323)  (which  in- 
volves very  little  additional  labor,  since  the  logarithms  of  sin  /9 
and  sin  JA  have  already  occurred  in  the  previous  computation), 
and  then  we  have  the  true  latitude 

<p'  =  <f>  +  A? 

If  we  wish  also  to  correct  the  hour  angle  r  found  by  this 
method,  we  have,  from  the  second  equation  of  (47)  applied  to 


268  LATITUDE. 

the  triangle  PTZ  (taking  b  and  c  to  denote  the  sides  P  T  and 
Z  T,  which  are  here  constant), 

cos  H  cos  A       -o 


AT  '  — 


COS 


ill  which  A  is  the  azimuth  of  the  point  T.  But  we  have  in  the 
triangle  PTZ 

cos  H  cos  A  ==.  sin  <p  cos  D  cos  r  —  cos  <p  sin  D 
Substituting  this  and  the  value  of  AP,  we  have 

AT  =  -      -  (tan  <p  cos  T  —  tan  D) 
sin  *  A 

and,  substituting  the  value  of  tan  D  (320), 

A'7     /  tan  3  \ 

AT  =  — I  tan  y  cos  T 

sin  £  A  \  cos  5  A  / 

When  this  correction  is  added  to  r,  we  have  the  value  that  would 
be  found  by  the  rigorous  formulae,  and  we  have  then  to  apply 
also  the  correction  x  according  to  (314).  In  the  present  case  we 
have,  by  (313), 

x  =  —  A<S  tan  d  tan  £  A 

and  the  complete  formulae  for  the  hour  angles  t  and  t'  become 

t   =  r  -f-  Ar  —  x  —  2  % 
f  =  T  +  AT  —  x  +  IJi 

Putting 

y  =  AT  —  x 

we  find,  by  substituting  the  above  values  of  AT  and  x, 

y  —  A<5 .  I  — — ^ — *—  )  (324) 

\     sin  i  A  tan  }  /  / 

and  then  we  have 


The  corrections  A^>  and  i/  are  computed  with  sufficient  accu- 
racy with  four-place  logarithms,  and,  therefore,  add  but  little  to 
the  labor  of  the  computation.  Nevertheless,  when  both  latitude 
and  time  are  required  with  the  greatest  precision,  it  will  be  pre- 
ferable to  compute  by  the  rigorous  formulae. 


BY   TWO    ALTITUDES.  269 

EXAMPLE.— 1856  March  10,  in  Lat.  24°  K,  Long.  30°  W., 
suppose  we  obtain  two  altitudes  of  the  sun  as  follows,  all  correc- 
tions being  applied :  find  the  latitude. 

At  app.  time      0*  30-  h    =  61°  11'  38".3   (d )  ==  —  3°  51'  52".8 

4  30  h'   =  18    46  35  .8   (<T)=  —  3    47  57  .4 

*  A  =     2*    0-  *  ^  H-  A')  =  39~59~    7  .1       '5  =  —  3~49  55  .1 

=  30°    0'  \  (h  —  h')  =  21    12  31  .3   A*  =      -f     V  57".7 

The  following  is  the  form  of  computation  by  the  formulae 
(316),  (317),  and  (318),  employed  by  BOWDITCH  in  his  Practical 
Navigator,  the  reciprocals  of  the  equations  (316)  and  of  the 
second  of  (317)  being  used  to  avoid  taking  arithmetical  comple- 
ments. This  form  is  convenient  when  the  tables  give  the  secants 
and  cosecants,  as  is  usual  in  nautical  works. 

cosec  £  A  0.301030 

sec  J                   0.000972 cosec  nl.  175024 

cosec  C             0.302002      cos     9.937854 cos  9.937854 

cos  $  (A  +  h')  9.884347      cosec  0.192065      D  =  —  4°  25'  21". S  cosec  nl.112878 
sin  l(h  —  h')  9.558428      sec      0.030459 

smp                 9.744777      cos      9.919829 cos  9.919829 

sec     0.080207       7=       33    45  38  .0 

D  -f  y  —       29    20  16  .7  sin  9.690161 

0  =       24°    2'  23".2  sin  9.609990 

If  the  approximate  latitude  had  not  been  given,  we  might  also 
have  taken  D  —  f  =  —  38°  10'  59".3,  and  then  we  should  have 
had 

cos  p    9.919829 

sin  (D  —  Y)  n9.791113 

<p  =  —  30°  55'  44".3          sin  <p  >/r  71  )942 

To  correct  the  first  value  of  the  latitude  for  the  change  of 
declination,  we  have,  by  (323), 

log  A 5  2.0708 
sin  /?  9.7448 
cosec  i  A  0.3010 
sec  c>  0.0394 
A?  =  —  143".2  log  A?  n^.im 

and  hence  the  true  latitude  is 

'  —  24°  2'  23".2  —  2'  23". 2  =  24°  0'  0" 


270  LATITUDE. 

which  agrees  exactly  with  the  value  computed  by  the  rigorous 
formulae. 

The  approximate  time  is  found  hy  the  last  equation  of  (318) 
with  but  oue  new  logarithm :  we  have 

sin  /?  9.744777 

cos?  9.960596 

r  ==  37°  28'  23"  sin  r  9.784181 

By  (324)  and  (325),  we  find 

log  A<S       2.0708  log  A<5  2.0708 

coscc  J  A  0.3010  cot  i  /I  0.2386 

tan  y         9.6494  tan  d  yi8.8259 

COST         9.8996       —  13".  7   wTi353 
-f  83".3         1.9208 

y  =  -f  83".3  —  (—  13".7)  =  -f  97" 

T  -f  y  =  37°  30'  0"  ==  2*  30"  0« 

t  =    0*  30-  0'          f  =  4*  SO"  0' 

which  are  perfectly  exact. 

182.  (D.)  Tivo  equal  altitudes  of  the  same  star,  or  of  the  sun. — This 
case  is  a  very  useful  one  in  practice  with  the  sextant  when  equal 
altitudes  have  been  taken  for  determining  the  time  by  the  method 
of  Art.  140.  By  putting  h'  =  h  in  the  formula  (317),  we  find 
sin  /?  =  0,  cos  £  =  1,  and  hence  (318)  gives  sin  <p  =  sin  (D  -f  p),  or 
We  have,  therefore,  for  this  case,  by  (320)  and  (321), 


tan  d  sin  h  sin  D 

tan  D  =  —  cos  y  =  - 

cos  $A  sin  d 


(326) 


which  is  the  method  of  Art,  164  applied  as  proposed  in  Art.  165. 
We  give  f  the  double  sign,  and  obtain  two  values  of  the  latitude, 
for  the  reasons  already  stated. 

The  time  will  be  most  conveniently  found  by  Art.  140.  The 
method  there  given  is,  however,  embraced  in  the  solution  of  the 
present  problem.  For,  since  we  here  have  sin  /?  =  0,  we  also 
have  T  =  0,  and  the  hour  angles  given  by  (325)  become 

t  =y-U 
t'  =  y  -h  i  A 


BY    TWO    ALTITUDES.  271 

the  mean  of  which  gives 

*  (*  +  O  -  y  =  o 

that  is,  — y  is  the  correction  of  the  mean  of  the  times  of  obser- 
vation to  obtain  the  time  of  apparent  noon  =  0*.  This  correction 
was  denoted  in  Art.  140  by  A7'0;  and  since  cos  r  =  0,  the  formula 
(324)  gives,  when  divided  by  15  to  reduce  it  to  seconds  of  time, 

tan  <p          A<5  tan  d 


—    — 

15  sin  £  A         15  tan  £  A 

which  is  identical  with  (262).    Thus  it  appears  that  (262)  is  but  a 
particular  case  of  the  formula  (324),  which  I  suppose  to  be  new. 
The  latitude  found  by  (326)  will  require  no  correction,  since 
sin  /9  =  0,  and  therefore  &<p  =  0. 

NOTE. — The  preceding  solutions  of  the  problem  of  finding  the  latitude  and 
time  by  two  altitudes  leave  nothing  to  be  desired  on  the  score  of  completeness  and 
accuracy ;  but  some  brief  approximative  methods,  used  only  by  navigators,  will  be 
treated  of  among  the  methods  of  finding  the  latitude  at  sea,  and  in  Chapter  VIII. 

183.  I  proceed  to  discuss  the  effect  of  errors  in  the  data  upon 
the  latitude  and  time  determined  by  two  altitudes,  and  hence 
also  the  conditions  most  favorable  to  accuracy. 

Errors  of  altitude. — Since  the  hour  angles  t  and  t'  are  connected 
by  the  relation  t'  =  t  -f-  ^,  the  errors  of  altitude  produce  the  same 
errors  in  both ;  for,  A  being  correct,  we  have  dt'  =  dt ;  and  for 
either  of  these  we  may  also  #ut  dr,  since  we  have,  in  the  second 
general  solution  of  Art.  178,  r  —  x  =  \(t  -f-  tf),  an4  x  is  not 
affected  by  errors  of  altitude.  Now,  we  have  the  general  relations 

sin  h  =  sin  <p  sin  d   -\-  cos  <p  cos  d  cos  t 
sin  hf  =  sin  y  sin  d'  -f  cos  <p  cos  d' cos  t' 

which,   by  differentiation  relatively  to   A,  ^,  and  /,  give   [see 
equations  (51)] 

dh  =  —  cos  A  d<p  —  cos  y>  sin  A  dr 
dhf  =  —  cos  A'dv  —  cos  ^  sin  A'dr 

in  which  A  and  A'  denote  the  azimuths  of  the  two  stars,  or  of 
the  same  star  at  the  two  observations. 


272  LATITUDE. 

Eliminating  dr  and  dtp  successively,  we  find 

sin  A'  sin  A 

d(p  — dh  -j dh' 

sin  (A  —  A)  sin  (A  —  A) 

cos  A'  cos  A         .... 

cos  </>  dr  = dA dA' 

sin  (4'  —  4)  sin  (A'  —  A) 


(328^ 


These  equations  show  that,  in  order  to  reduce  the  effect  of  error? 
of  altitude  as  much  as  possible,  we  must  make  sin  ( A'  —  A)  a? 
great  as  possible,  and  hence  Af  —  A,  the  difference  of  the  azi- 
muths, should  be  as  nearly  90°  as  possible. 
If  we  suppose  A'  —  A  =  90°,  we  have 


d(p  =  —  sin  A' dh  -f  mnAdh' 
cos  <f>  dr  =       cos  A'  dh  —  cos  A  dh' 


and,  under  the  same  supposition,  if  one  of  the  altitudes  is  near 
the  meridian  the  other  will  be  near  the  prime  vertical.  If,  then, 
the  altitude  h  is  near  the  meridian,  sin  A  will  be  small  while 
sin  A'  is  nearly  unity,  and  the  error  dtp  will  depend  chiefly  on 
the  term  sin  A'dh.  At  the  same  time,  cos  A  will  be  nearly  unity 
and  cos  A'  small,  so  that  the  error  dr  will  depend  chiefly  on  the 
term  cosAdh'.  Hence  the  accuracy  of  the  resulting  latitude  wTill 
depend  chiefly  upon  that  of  the  altitude  near  the  meridian ;  and 
the  accuracy  of  the  time  chiefly  upon  that  of  the  altitude  near 
the  prime  vertical. 

If  the  observations  are  taken  upon  the  same  star  observed  at 
equal  distances  from  the  meridian,  we  have  A1  —  —  A,  and  the 
general  expressions  (328)  become 

dh  +  dh' 
d<p  =  —  - 

2  cos  A 

dh  —  dh' 

COS  <p  dr  =  —  • : • 

2  sin  A 

The  most  favorable  condition  for  determining  both  latitude 
and  time  from  equal  altitudes  is  sin  A  =  cos  A,  or  A  =  45°. 

Errors  in  the  observed  clock  times. — An  error  in  the  observed 
time  may  be  resolved  into  an  error  of  altitude  ;  for  if  we  say  that 
dT  is  the  error  of  T  at  the  observation  of  the  altitude  h.  it 


BY   TWO    ALTITUDES.  273 

amounts  to  saying  either  that  the  time  T—  dT  corresponds  to 
the  altitude  A,  or  that  T  corresponds  to  the  altitude  h  -f  dh,  dh 
being  the  increase  of  altitude  in  the  interval  dT.  We  may, 
therefore,  consider  the  time  T  as  correctly  observed  while  h  is  in 
error  by  the  quantity  —  dh.  Conversely,  we  may  assume  that 
the  altitudes  are  correct  while  the  times  are  erroneous.  The 
discussion  of  the  errors  under  the  latter  form,  while  it  can  lead 
to  no  new  results,  is,  nevertheless,  sufficiently  interesting.  We 
have,  from  the  formula  (304), 


The  general  equations  (327),  upon  the  supposition  that  h  and  A' 
are  correct,  give 

0  =  —  cos  A  dtp  —  cos  <p  sin  A  dt 

0  =  —  cos  A'  d<p  —  cos  y  sin  A'  (dt  -{-  dk)  * 

where  we  put  dt  +  dl  for  dt'  ^  since  t'  =  t  -f-  L     Eliminating  dt,  we 

find 

cos  y  sin  A'  sin  A 


sin  (A'  —  A) 
Eliminating  dp, 


dl  (329) 


and  similarly 


sin  A'  cos  A   ,, 
dt  = d). 

sin  (A'  —  A) 

sin  A  cos  Af    ,. 

dt  — -ax 

sin  (A'  — A) 


But  we  have  from  the  formula  r  —  x  =  J  (t  +  tr) 

dr=  J  (dt  +  <#') 
and  hence 

Sin  (.yl    -f-  .AJ      <ZA  /OOA\ 

-sin^'-^'T 

If  we  denote  the  clock  correction  at  the  time  T  by  #,  we  shall 
have 

*  =  *  -f  a  —  T 
and 

VOL.  I.— IS 


274  LATITUDE. 

or,  if  we  deduce  &  from  the  second  observation,  the  rate  being 
supposed  correct, 

d&  =  dt'  —  dT' 
The  mean  is 


Substituting  the  value  of  dr  and  also  dX  =  dT1 —  dT,  we  find, 
after  reduction, 

sin  A  cos  A'  dT_  **A'&*A 

sin  (A'  —  A}  sin  (A'  —  A) 

The  conclusions  above  obtained  as  to  the  conditions  favorable  to 
the  accurate  determination  of  either  the  latitude  or  the  time  are, 
evidently,  confirmed  by  the  equations  (329)  and  (331).  In  addi- 
tion, we  may  observe  that  if  d Tf  =  dT,  we  have  dtp  =  0  and 
d&  =  dT:  a  constant  error  in  noting  the  clock  time  produces  no 
error  in  the  latitude,,  but  affects  the  clock  correction  by  its  whole 
amount. 

Errors  of  declination. — These  errors  may  also  be  resolved  into 
errors  of  altitude.  By  differentiating  the  equations  (327)  rela- 
tively to  h  and  d,  we  find 

dh  =  cos  qdd,       dhr  —  cos  q' dd' 

in  which  q  and  q'  are  the  parallactic  angles  at  the  two  observa- 
tions respectively.  If  these  values  are  substituted  in  (328),  the 
resulting  values  of  d<p  and  dr  will  be  the  corrections  required  in 
the  latitude  and  hour  angle  for  the  errors  dd  and  ddf  ;*  and,  de- 
noting these  corrections  by  A^>  and  Ar,  we  have 

sin  A'  cos  q  sin  A  cos  q'       , 

sin  (A  —  A}  sin  (A'  —  A} 

(332) 

cos  A'  cos  q  cos  A  cos  q' 

-L.  dd —Z-  dd' 

sin  (A'  —  A)  sin  (A1  —  A) 

If  the  observation  h  is  on  the  meridian,  and  hf  on  the  prime 
vertical,  we  have  A^>  =  —  dd ;  and  in  the  same  case  we  have,  by 

*  The  error  of  a  quantity  and  the  correction  for  this  error  are  too  frequently  con- 
founded. They  are  numerically  equal,  but  have  opposite  signs.  If  a  should  be 
,a  —  x,  it  is  too  great  by  z ;  its  error  is  -}-  x  /  but  the  correction  to  reduce  it  to  its 
true  value  is  —  z. 


BY   TWO    ALTITUDES.  275 

(328),  d(f>  =  -j-  dh,  and  the  total  correction  of  the  latitude 
=  dh  —  dd,  precisely  the  same  as  if  the  meridian  observation 
were  the  only  one.  Hence  there  is  no  advantage  in  combining 
an  observation  on  the  meridian  with  another  remote  from  it,  in 
the  determination  of  latitude :  a  simple  meridian  observation, 
or,  still  better,  a  series  of  circummeridian  observations,  is  always 
preferable  if  the  time  is  approximately  known. 

When  the  sun  is  observed  and  the  mean  declination  is  em- 
ployed, putting  A£  =  J  (d'  —  d)j  we  have  do  =  A£,  dd'  =  —  A£, 
and,  by  (332), 

sin  A'  cos  q  -f  sin  A  cos  q' 

sin  (A'  —  A) 
which,  by  substituting 

sin  a'  cos  d  sin  a  cos  d 

em  A'  =  ——i sin  A  = " 

cos  <p  cos  <f> 

becomes 

si"  (q'  +  q)  cos  d 

&<f>  =  — •  Ao  ^oooj 

sin  (A  —  A)  cos  </> 

This  is  but  another  expression  of  the  correction  (323). 

If,  when  the  sun  is  observed,  instead  of  employing  the  mean 
declination  we  employ  the  declination  belonging  to  the  greater 
altitude,  which  we  may  suppose  to  be  A,  we  shall  have  dd  =  0, 
do'  =  —  2  A£  ;  and,  denoting  the  correction  of  the  latitude  in 
this  case  by  A'^>  we  have,  by  (332), 

,  2  sin  A  cos  q'  2  sin  q  cos  q'  cos  d 

sin  (A'  —  A)  sin  (A'  —  A)  cos  <f> 

Under  what  conditions  will  A'^  be  numerically  less  than  A^  ? 

First.  If  both  observations  are  on  the  same  side  of  the 
meridian,  the  condition  A'$P  <  A^>  gives 

2  sin  q  cos  q'  <  sin  (q'  -j-  q) 
or 

2  sin  q  cos  q'  <  sin  q'  cos  q  -f-  cos  q'  sin  q 
whence 

tan  q  <  tan  q' 

which  condition  is  always  satisfied  when  h  is  the  greater  altitude. 
Secondly.  If  the   observations   are   on   different   sides  of  the 


276  LATITUDE. 

meridian,  q  and  q'  will  have  opposite  signs,  and  we  shall  have, 
numerically,  sin  (qf  —  q)  instead  of  sin  (qf  -j-  q).  The  condition 
&f<p  <C  A^>,  then,  requires  that 

2  sin  q  cos  q'  <  sin  q'  cos  <?  —  cos  q'  sin  £ 
or 

tan  q  <  |  tan  ^ 

Therefore  A  ^'  will  be  greater  than  &y  only  when  the  observa- 
tions are  on  opposite  sides  of  the  meridian  and  tan  q  >  J  tan  qr. 
In  cases  where  an  approximate  result  suffices,  as  at  sea,  and  the 
correction  A^  is  omitted  to  save  computation,  it  will  be  expedient 
to  employ  the  declination  at  the  greater  altitude,  except  in  the 
single  case  just  mentioned.*  But  to  distinguish  this  case  with 
accuracy  we  should  have  to  compute  the  angles  q  and  q1  ';  and 
therefore  an  approximate  criterion  must  suffice.  Since  the 
parallactic  angles  increase  with  the  hour  angles,  we  may  substi- 
tute for  the  condition  tan  q  >  J  tan  qf  the  more  simple  one 
t  >  J  £',  which  gives 


or  (t  and  t'  being  only  the  numerical  values  of  the  hour  angles) 


Hence  we  derive  this  very  simple  practical  rule  :  employ  the  sun's 
declination  at  the  greater  altitude,  except  when  the  time  of  this  altitude 
is  farther  from  noon  than  the  middle  time,  in  which  case  employ  the 
mean  declination. 

The  greatest  error  committed  under  this  rule  is  (nearly)  the 
value  of  A^  in  (323),  when  T  =  t.  But  since  in  this  case  3£  =  t'  , 
and  t  -f-  1'  =  /,  we  have  T  =  J  ^,  and  therefore  sin  /*  =  cos  tp  sin  r 
=  cos  <p  sin  J  L  This  reduces'  (323)  to  &<p  =  -  J  AO  sec  J  L 
Since  ^  will  seldom  exceed  6\  A^  will  not  exceed  3r,  and  the 
maximum  error  will  not  exceed  1'.6.  In  most  cases  the  error 
will  be  under  1',  a  degree  of  approximation  usually  quite  suffi^ 
cient  at  sea.  Nevertheless,  the  computation  of  the  correction 
&(p  by  our  formula  (323)  is  so  simple  that  the  careful  navigator 

*  BOWTMTCII  nnd  navigators  generally  employ  in  all  cases  the  mean  declination; 
but  the  above  discussion  proves  that,  if  the  cases  are  not  to  be  distinguished,  it  will 
be  better  always  to  employ  the  declination  at  the  greater  altitude. 


BY    TWO    EQUAL   ALTITUDES.  277 

will  prefer  to  employ  the  mean  declination  and  to  obtain  the 
exact  result  by  applying  this  correction  in  all  cases. 

SIXTH    METHOD. — 'BY   TWO    ALTITUDES    OF    THE    SAME    OR    DIFFERENT 
STARS,  WITH    THE   DIFFERENCE    OF   THEIR   AZIMUTHS. 

184.  Instead  of  noting  the  times  corresponding  to  the  observed 
altitudes,  we  may  observe  the  azimuths,  both  altitude  and  azi- 
muth  being   obtained   at   once   by  the  Altitude  and  Azjmuth 
Instrument  or  the  Universal  Instrument.     The  instrument  gives 
the  difference  of  azimuths  =  L     The  computation  of  the  latitude 
and  the  absolute  azimuth  A  of  one  of  the  stars  may  then  be 
performed  by  the  formulae  of  the  preceding  method,  only  inter- 
changing altitudes  and  declinations  and  putting  180°  —  A  for  /. 
"When  A  has  been  found,  we  may  also  find  t  by  the  usual  methods. 

SEVENTH    METHOD. — BY   TWO    DIFFERENT    STARS    OBSERVED    AT    THE 
SAME   ALTITUDE    WHEN    THE    TIME    IS    GIVEN. 

185.  By  this  method  the  latitude  is  found  from  the  declinations 
of  the  two  stars  and  their  hotjr  angles  deduced  from  the  times 
of  observation,  without  employing  the  altitude  itself ,  so  that  the  result 
is  free  from  constant  errors  (of  graduation,  &c.)  of  the  instrument 
with  which  the  altitude  is  observed.     Let 

0,  0'  =  the  sidereal  times  of  the  observations, 
o,  a!    =  the  right  ascensions  of  the  stars, 
d,  df   =  the  declinations  li          " 

t,   t'    —  the  hour  angles  "  " 

h    =  the  altitude  of  either  star, 

tp    =  the  latitude ; 

then,  the  hour  angles  being  found  by  the  equations 

t  =  0  —  a  t'  =  0'  —  a' 

we  have 

gin  h  =  sin  <p  sin  d  -f-  cos  <p  cos  d  cos  t 
sin  h  =  sin  <p  sin  d'  -f-  cos  <p  cos  d'  cos  if 

From  the  difference  of  these  we  deduce 

tan  <p  (sin  d'  —  sin  d)  =  cos  d  cos  i  —  cos  dr  cos  tf 

=  $  (cos  d  —  cos  d')  (cos  t  -f-  cos  ?') 
-f  4  (cos  5  -f-  cos  <*')  (cos  t  —  cos  f) 


278  LATITUDE. 

and,  by  resolving  the  quantities  in  parentheses  into  their  factors, 

tan  <f>  =  tan  J  (<5'  -f  d)  cos  i  (*'  -f  f)  cos  J  (f  —  f) 
-|-  cot  J  (<5'  —  £)  sin  *  (f  +  0  sin  J  (f  —  Q 

If  now  we  put 

m  sin  M=  sin  £  (f  —  t)  cot  *  ((5'  —  6)  1 

m  cos  M=  cos  i  (f  —  t)  tan  *  (3'  -f  8)  j  V 

we  have 

tan  ?>  =  m  cos  [J  (f  -f  f)  —  J/]  (335) 

The  equations  (334)  determine  m  and  M,  and  then  the  latitude  is 
found  by  (335)  in  a  very  simple  manner. 

It  is  important  to  determine  the  conditions  which  must  govern 
the  selection  of  the  stars  and  the  time  at  which  they  are  to  be 
observed.  For  this  purpose  we  differentiate  the  above  expres- 
sions for  sin  h  relatively  to  <p  and  t.  The  error  in  the  hour  angles 
is  composed  of  the  error  of  observation  and  the  error  of  the  clock 
correction.  If  we  put 

T,  T'  =  the  observed  (sidereal)  clock  time, 
A  T  =  the  clock  correction, 
dT  =  the  rate  of  the  clock  in  a  unit  of  clock  time, 

we  shall  have 

t  =  T  -\-  &T—  a,       f  =  T'  -f-  AT7  -f  dT(Tf  _  T)  —  a' 

Differentiating  these,  assuming  that  the  rate  of  the  clock  is  suffi- 
ciently well  known,  we  have 

dt  =  dT+  d*T,       dtf  =  dT'  +  d±T 

in  which  dT9  dT'  are  the  errors  in  the  observed  times,  ande?A  T 
the  error  in  the  clock  correction.     The  differential  equations  are 

then 

dh  =  —  cos  Ad<p  —  cos  <p  sin  AdT  —  cos  <p  sin  A  d&T 
dh  =  —  cos  A' d(f  —  cos  y  sin  A'dT'  —  cos  <p  sin  A'd&T 

in  which  A  and  Af  are  the  azimuths  of  the  stars.    The  difference 
of  these  equations  gives 

_^_==__          8in^         dT\          sin;!'         dT,      KmA'  —  »\n 
cosy?          cos  A — cos  ^4'  cos  A  —  cos  A'  cos  A'  —  cos  A 


BY   TWO    EQUAL   ALTITUDES.  279 

The  denominator  cos  A  —  cos  A'  is  a  maximum  when  one  of 
the  azimuths  is  zero  and  the  other  180°,  that  is,  when  one  of  the 
stars  is  south  and  the  other  north  of  the  observer.  To  satisfy 
this  condition  as  nearly  as  possible,  two  stars  are  to  be  selected 
the  mean  of  whose  declinations  is  nearly  equal  to  the  latitude, 
and  the  common  altitude  at  which  they  are  to  be  observed  will 
be  equal  to  or  but  little  less  than  the  meridian  altitude  of  the 
star  which  culminates  farthest  from  the  zenith.  It  is  desirable, 
also,  that  the  difference  of  right  ascensions  should  not  be  great. 

The  coefficient  of  d^Tis  equal  to  —  cot  %(Af  -f  A),  which  is 
zero  when  %(Af  +  A)  is  90°  or  270° :  hence,  when  the  observa- 
tions are  equally  distant  from  the  prime  vertical,  one  north  and 
the  other  south,  small  errors  in  the  clock  correction  have  no 
sensible  effect. 

When  the  latitude  has  been  found  by  this  method,  we  may 
determine  the  whole  error  of  the  instrument  with  which  the 
altitude  is  observed;  for  either  of  the  fundamental  equations 
will  give  the  true  altitude,  which  increased  by  the  refraction  will 
be  that  which  a  perfect  instrument  would  give. 

186.  With  the  zenith  telescope  (see  Vol.  II.)  the  two  stars 
maybe  observed  at  nearly  the  same  zenith  distance,  the  small 
difference  of  zenith  distance  being  determined  by  the  level  and 
the  micrometer.  The  preceding  method  may  still  be  used  by 
correcting  the  time  of  one  of  the  observations.  If  at  the  ob- 
served times  T,  Tr  the  zenith  distances  are  £  and  f ',  the  times 
when  the  same  altitudes  would  be  observed  are  either 

T     and     T'  + 


cos  y  sin  A' 
or, 

T  -| ~  * —      and     T' 

cos  ?>  sin  A 

where  £'  —  £  is  given  directly  by  the  instrument.  With  the 
hour  angles  deduced  from  these  times  we  may  then  proceed  by 
(334)  and  (335). 

But  it  will  be  still  better  to  compute  an  approximate  latitude 
by  the  formulae  (334)  and  (335),  employing  the  actually  observed 
times,  and  then  to  correct  this  latitude  for  the  difference  of 
zenith  distance. 


280  LATITUDE. 

By  differentiating  the  formula 

tan  if  (sin  3'  —  sin  <5)  =  cos  8  cos  t  —  cos  3'  cos  f 
relatively  to  <p  and  tf,  we  have 

sec2  y  (sin  d'  —  sin  <S)  dtp  —  cos  8'  sin  f  dt'  =  sin  C  sin  ^1' 
Hence,  substituting 


df  =  dT'  == 


we  find 


eos  <p  sin  A 
(C  —  C')  sin  C  cos  <p 


(336) 


sin  i  (£'  —  8)  cos  £  (#  -f  8} 
and  the  true  latitude  will  be  <p  -f  <fy. 

EIGHTH    METHOD. — BY  THREE  OR  MORE  DIFFERENT    STARS    OBSERVED 
AT   THE    SAME   ALTITUDE   WHEN   THE  TIME    IS    NOT    GIVEN. 

187.  To  find  both  the  latitude  and  the  clock  correction  from  the  clock 
times  when  three  different  stars  arrive  at  the  same  altitude. 

As  in  the  preceding  method,  we  do  not  employ  the  common 
altitude  itself;  and,  as  we  have  one  more  observation,  we  can  de- 
termine the  time  as  well  as  the  latitude. 

Let  S,  S',  #",  Fig.  26,  be  the  three  points  of  the  celestial 
sphere,  equidistant  from  the  zenith  Z,  at  which 
the  stars  are  observed.  Let 

T,  T',  T"  —  the  clock  times  of  the  observations, 

A  T  =  the  clock  correction  to  sidereal  time  at 

the  time  T, 
8T  =  the   rate    of  the  clock  in  a  unit   of 

clock  time, 

a,  a',  a"  =  the  right  ascensions  of  the  stars, 
8,  8',  8'  =  the  declinations          "  " 

t,  1f,  t"  =  the  hour  angles          u  " 

h  =  the  altitude, 
<p  =  the  latitude. 

X  =  f  —  t  =  SPS', 
X'  =  t"  —  t  =  SPS"', 

then,  since  the  sidereal  times  of  the  observations  are 


Also,  let 


BY   THREE    EQUAL    ALTITUDES.  281  * 


0    =   T  '  + 

0'  =  T'  +-&T+  8T(Tf  —T) 
0"  =  T"  +  AT+  dT(T"—T) 

and  the  hour  angles  are 

<=0—  a,        f=0'—  «/,        r=0"—  a", 

we  have 

A   =   T7'  —  T  +  3T  (Tf  —  T)  —  (a'    —  a) 
X  =   Tn_  T  +  8T^Tn__  T)  _  (a"  _  a) 

The  angles  A  and  A'  are  thus  found  from  the  observed  clock 
times,  the  clock  rate,  and  the  right  ascensions  of  the  stars.  The 
hour  angles  of  the  stars  being  t,  t  +  ^,  and  t  -f-  A',  we  have 

sin  h  =  sin  $0  sin  <S    -j-  cos  ^  cos  d    cos  f 

sin  h  =  sin  ^>  sin  dr  -j-  cos  ^»  cos  d'  cos  (f  -j-  A) 

sin  h  =  sin  ^  sin  d"  -\-  cos  ^  cos  d"  cos  (^  -j-  A') 

Subtracting  the  first  of  these  from  the  second,  we  have  an  equa- 
tion of  the  same  forjn  as  that  treated  in  Art.  185,  only  here  we 
have  t  +  A  instead  of  t'  ;  and  hence,  by  (334),  we  have 


m  sin  M  =  sin  J  A  cot  \  (<$'  — 
mcos  Jf  =cos^A  tanj<5'- 


"I 
/ 

and  putting 

N=%  l  —  M  (338) 

we  have,  by  (335), 

tan  <f>  =  m  cos  (t  -f-  JV)  (339) 

In  the  same  manner,  from  the  first  and  third  observations  we 
have 

m'  sin  M'  =  sin  \  A'  cot  1(3"  —  d)  1     r340 

m'  cos  Jf  '  r=  cos  }  /'  tan  }  (d"  +  d)  J 

JV'  =  i  A'  —  Jf  '  (341) 

tan  <p  =  m!  cos  (f  +  JV')  (342) 

The  problem  is  then  reduced  to  the  solution  of  the  two  equa- 
tions (339)  and  (342),  involving  the  two  unknown  quantities 
y  and  t.  If  we  put 

k  cos  (t  -|-  JV)  = 


282  LATITUDE. 

there  follows  also 

k  cos  ft  +  JV)  =  — 
m' 

and  these  two  equations  are  of  the  form  treated  of  in  PL  Trio:. 
Art.  179,  according  to  which,  if  the  auxiliary  &  is  determined  by 
the  condition 

tan  #  =  —  (343) 

m' 

we  shall  have 

tan  p  -f-  }  (JV+  JV')]  =  tan  (45°  —  0)  cot  }  (^'  —  JV}       (344) 


which  determines  2,  from  which  the  clock  correction  is  found  by 
the  formula 

&T=  a  -f  t  —  T 

The  latitude  is  then  found  from  either  (339)  or  (342).* 
To  determine  the  conditions  which  shall  govern  the  selection 
of  the  stars,  we  have,  as  in  Art.  185, 

dh  =  —  cos  A  dip  —  cos  <p  sin  A  dT   —  cos  <p  sin  A  d  A  T 
dh  =  —  cos  A'  d(p  —  cos  <f>  sin  A'  dT'  -—  cos  y  sin  A' 
dh  =  —  cos  A"  dy  —  cos  <p  sin  A"  dT"  —  cos  <p  sin  A" 


By  the  elimination  of  d&  T,  we  deduce  the  following: 

(sin  A  —  sin  A'  )  dh  =  sin  (A1  —  A  )  dq  —  cos  0  sin  A'  sin  A  (dT'  —  dT  ) 
(sin  A'  —  sin  A")  dh  =  sin  (A"  —  A'  )  <fy  —  cos  0  sin  ^L"  sin  ^1'  (rfT7"  —  rfT"  ) 
(sin  A"  —  sin  A  )  dh  —  sin  (.4  —  A")  efy>  —  cos  0  sin  .4  sin  A"  (dT  —  dT") 

Adding  these  three  equations  together,  and  putting 

2,K=  sin  (A'  —  A)  +  sin  (A"  —  A')  +  sin  (A  —  A") 


we  find 


sin  A  (sin  A"  —  sin  A'}  sin  A'  (sin  ^4  —  sin  A") 


2K  2K 


,     sin  A"  (sin  ^'  —  sin  A) 


By  eliminating  ffy>  from  the  same  three  equations,  we  shall  find 

*  This  simple  and  elegant  solution  is  due  to  GAUSS,  Monatliche  Correspondenz,  Vol. 
XVIII.  p.  287. 


BY    THREE    EQUAL    ALTITUDES.  283 

sin  . 4  (cos  A'-  cos  A")  ^       sin  A'  (cos  A"  -  cos  ^)  ^ 
2^T  2Ji: 

sin  ^4"  (cos  A  —  cos  A') 

The  denominator  2  7T is  a  maximum  when  the  three  differences 
of  azimuth  are  each  120°,*  which  is  therefore  the  most  favorable 
condition  for  determining  both  the  latitude  and  the  time.  In 
general,  small  differences  of  azimuth  are  to  be  avoided. 
%  GAUSS  adds  the  following  important  practical  remarks.  It  is 
clear  that  stars  whose  altitude  varies  slowly  are  quite  as  available 
as  those  which  rise  or  fall  rapidly ;  for  the  essential  condition  is 
not  so  much  that  the  precise  instant  when  the  star  reaches  a 
supposed  place  should  be  noted,  as  that  at  the  time  which  is 
noted  the  star  should  not  be  sensibly  distant  from  that  place. 
We  may,  therefore,  without  scruple  select  one  of  the  stars  near 
its  culmination,  or  the  Pole  star  at  any  time,  and  we  can  then 
easily  satisfy  the  above  condition.  Moreover,  at  least  one  of  the 
stars  will  always  change  its  altitude  rapidly  when  the  condition 
of  widely  different  azimuths  is  satisfied. 

The  stars  proper  t6  be  observed  may  be  readily  selected  with 
the  aid  of  an  artificial  globe,  and  in  general  so  that  the  intervals 
of  time  between  the  observations  shall  be  so  small  that  irregu- 
larities of  the  clock  or  an  error  in  the  assumed  rate  shall  not 
have  any  sensible  influence. 

Having  selected  the  stars,  the  clock  times  when  they  severally 
arrive  at  the  assumed  altitude  are  to  be  computed  in  advance 
within  a  minute  or  two,  so  that  the  observer  may  be  ready  for 
each  observation  at  the  proper  time.  This  is  quickly  done  with 
four-place  logarithms  by  the  formula  (267),  in  which  <p  and  £ 
will  have  the  same  values  for  the  three  stars. 

*  For  by  putting  a  —  A'  —  A,  a'  =  A"  —  A',  we  have 

2  K  =  sin  a  -f  sin  a'  —  sin  (a  -f  a') 

and,  differentiating  with  reference  to  a  and  a',  the  conditions  of  maximum  or  mini- 
mum are 

cos  a  —  cos  (a  -(-  a')  =  0 

cos  a'  —  cos  (a  +  a')  =  0 

which  give  either  a  =  a'  =  0  or  a  =  a'  —  120°;  and  the  latter  evidently  belongs  to 
the  case  of  maximum. 


284  LATITUDE. 


If  it  is  desired  to  compute  the  differential  formulae,  the,  follow- 
ing form  will  be  convenient.     We  have 

K  =  —  2  sin  %(A  —  A)  sin  }  (A"  —  A)  sin  }  (A  —  A") 

d<p          _  sin  A  cos  \  (A"  -f  .4')  sin  $  (JT  —  yl')      _ 
15  cos  <f>  K 

sin  4'  cos  \(A  -f  A")  sin  £  ( A  —  A")      T, 

~K~  * 


Bin  A"  cos  j  (A'  +  A)  sin  j  (A'  -  4)   ^,, 


d±T=  8in  A  Sin  *  ^"  "^"  A'^  Si"  *  ^"    -^0   dT 

K 

sin  ^1'  sin  $  (A  4-  A"}  sin  £  f  A  —  ^4") 

+ T4 ' dT 

— |—  __ — •  .  (j,  j[ 

K 

where  dtp  is  divided  by  15,  since  it  will  be  expressed  in  seconds 
of  arc,  while  d  T,  dT',  and  dT"  are  in  seconds  of  time.  If  we 
first  compute  the  coefficients  of  the  value  of  d&T,  those  of 
d(p  will  be  found  by  multiplying  the  former  respectively  by 
cot  J  (Af  -\-  A"),  cot  J  ( A  -\-  A"},  and  cot  J  ( A'  -f  A),  and  also  by 
15  cos  (p.  It  is  well  to  remark,  also,  for  the  purpose  of  verifica- 
tion, that  the  sum  of  the  three  coefficients  in  the  formula  for  d(p 
must  be  =  0,  and  the  sum  of  those  in  the  formula  for  d  A  T  must 
be  =  —  1. 

The  substitution  of  dl  for  dT'—dT,  and  dl'  for  dT"  —  dT, 
will  reduce  the  above  expressions  to  a  more  simple  form,  which 
I  leave  to  the  reader. 

EXAMPLE. — To  illustrate  the  above  method,  GAUSS  took  the 
following  observations,  with  a  sextant  and  mercurial  horizon,  at 
Gottingen,  August  27,  1808.  The  double  altitude  on  the  sextant 
was  105°  18'  55".  The  time  was  noted  by  a  sidereal  clock 
whose  rate  was  so  small  as  not  to  require  notice. 


BY   THREE    EQUAL    ALTITUDES. 


285 


a  Andromedce  T  =  21*  33*  26- 
a  Ursa  Minoris  T'  =  21  47  30 
*Lyrce  2  "=22  5  21 

The  apparent  places  of  the  stars  were  as  follows : 


a  Andromedce      a  =  23*  58™  33«.33 
a  Ursce  Minoris  a'  =    0   55     4 .70 


a  Lyrce 
Hence  we  find 


18   30    28.96 


3  =  28°  2'  14".8 
3'  =  88  17  5  .7 
<S"  =  38  37  6  .6 


$X  =  —    5°18'25".28 
(<5'  —  3)=       30      7  25  .45 

(3f  -j-  <S)  —       58      9  40  .25 

'—<5)  0.2363973 
nS.9661070 
W9.2025043 


log  cot 
log  sin  £  A 
log  m  sin  J^f 


log  tan  *( 
log  cos  i  A 
log  m  cos 


0.2069331 
9.9981343 
0.2050674 


*  A'  iB±  44° 

__  ^)  =    5 

}  («"  -f  <5)  =  33 

log  cot  i  (*"  —  <5) 
log  sin  K 
log  w'  sin  Jf' 


69'  55".28 
17  25  .90 
19  40  .70 

1.0333869 
9.8494751 

0.8828620 


log  tan  £(<$"  +  (5)  9.8179461 
log  cos  J  A'  9.8494949 

log  m'  cos  Jf  '         9.6674410 


log  tan  M 
log  cos  J/ 
log  m 


w8.9974369 
9.9978645 
0.2072029 


log  tan  M' 
log  sin  M' 
log  m' 


1.2154210 
9.9991963 
0.8836657 


M  =  — 


5°  40'  37".96 
0    22  12  .68 


=     86°  30'  55".07 
'=  —  4:l    30  59  .79 


=      11°53'41".28  log  *  =  log 


tan 


9.3235372 


45°- 

-#=      33 

6 

18 

.72  log 

tan  (45°  —  *) 

9.8142617 

V7'—  J\T)  =  _20 

56 

36 

.24  log  cot  \ 

(W 

-Jf) 

wO.4171063 

V"  +  .    ' 

nrj=£i-59 

35 

14 

.71  log 

tan  [i 

?  +  i 

./  jyi    |       !\7"\ 

]  nO.2313680 

V  -L-  - 

AT)  =  __20 

34 

23 

.56 

^-=  —  39 

0 

51 

.15  =  - 

-    2* 

36- 

3-.41 

a  — 

23 

58 

33.33 

f 

+ 

a  = 

~  v9  —  ^ 

21 

22 

29.92 

T  = 

21 

33 

26. 

Clock  correct 

ion 

*T  = 

— 

10 

56.08 

Then,  to  find  the  latitude,  we  have 


286 


LATITUDE. 


t  +  N=  —  38°38'38".47 
log  cos  (t  -f  N)       9.8926738 


t  -f  N'  =  —  80°  31'  50".94 
log  cos  (t  -f  JV')      9.2162110 


log  m 
log  tan 


0.2072029 
0.0998767 


log  m' 
log  tan  <p 


0.8836657 
0.0998767 


<p  =  51°  31'  51".46 

If  with  these  results  we  compute  the  true  altitude  of  the 
stars,  we  find  from  each  h  =  52°  37'  21". 2.  The  refraction  was 
42".7,  and  hence  the  apparent  altitude  =  52°  38'  3".9.  The 
double  altitude  observed  was,  therefore,  105°  16'  7".8.  The 
index  correction  of  the  sextant  was  —  3'  30",  and  hence  the 
double  altitude  given  by  the  instrument  was  105°  15'  25", 
which  was,  consequently,  too  small  by  43". 

To  compute  the  differential  equations,  we  find 

A  =  293°  45'.2      A'  =  182°  9M      A"  =  90°  17'.9 

and  hence 

df  =  +  3.808  dT—  0.288  dT'  —  3.519  dT" 
d*T=  —  0.391  dT—  0.007  dT9  —  0.602  dT" 

by  which  we  see  that  an  error  of  one  second  in  each  of  the 
times  would  produce  at  the  most  but  7". 6  error  in  the  latitude, 
and  one  second  in  the  clock  correction. 

188.  Solution  of  the  preceding  problem  by  CAGNOLI'S  formulae. — 
After  GAUSS  had  published  the  solution  above  given,  he  was 
himself  the  first  to  observe*  that  CAGNOLI'S  formulae  for  the 
solution  of  a  very  different  problemf  might  be  applied  directly 
to  this. 

When  the  altitude  is  also  computed,  CAGNOLI'S  formulae  have 
slightly  the  advantage  over  those  of  GAUSS.  To 
deduce  them,  let  <?,  <?',  <?"  be  the  parallactic  angles 
at  the  three  stars,  or  (Fig.  26)  let 

q  =  PSZ,         q '  =  PS'Z,         q"  =  PS"Z, 


Fig.  26.  (bis). 


and  also  put 


Q    = 
Q'  = 


'  —  PS'S"} 
(PS"S  —  PSS") 

-PSS') 


*  Monatliche  Correspondenz,  Vol.  XIX.  p.  87. 

f  Namely,  that  of  determining,  from  three  heliocentric  places  of  a  solar  spot,  the 
position  of  the   sun's  equator,  and   the   declination  of  the   spot. — See  CAGNOLI'S 

Trigonometric,  p.  488. 


BY    THREE    EQUAL   ALTITUDES. 


287 


then,  since  ZSS',  ZS'S",  and  ZSS  "  are  isosceles  triangles,  we 
have 


whence 


q'  +  PS'S"  =  PS"Sf  —  q" 
q  +  PSS"    =  PS"S  —  q" 

0   +^  =  2Q» 
j'  +  q    =  2Q' 

t  +  <f  +  f  =  Q  +  Q'  +  Q" 

q  =  -  Q  4-  C'  +  C" 
g'  =      C  -  C'  +  Q" 

q"=        Q+  Q'-  q" 


(345) 


Now,  §,  §r,  §/r  are  found  from  the  triangles  PS"S',  PS"S, 
and  PS'S,  by  NAPIER'S  Analogies  (Sph.  Trig.  Art.  73),  thus : 


cos  J  (a"  +  <J') 


cos  i  (*"  +  f) 


cos 


where  ^,  A;  are  the  angles  at  the  pole  found  as  in  the  preceding 
article.     With  these  values  of  §,  Q',  Q",  those  of  q,  q',  and  q" 
become  known  by  (345). 
We  have  also 


whence 


and  from  this 


or 


cos  <f>  sin  (t  -|-  A)  =  cos  h  sin  q' 
cos  <p  sin  t  =  cos  A  sin  q 


sin  (f  -j- 


sn 


sn 


sin  q 


sin  (f  -f-  A)  -f-  sin  f  _  sin  q'  -f-  sin  </ 
sin  (t  -f-  A)  —  sin  £        sin  #'  —  sin  q 


tan 


tan  i  A 


tan  J     '  — 


288  LATITUDE. 

Substituting  the  values  of  q  and  qf  in  terms  of  §,  this  gives 

tan  (t  -f-  JA)  =  tan  U  tan  Q"  cot  (§  —  Qf) 
or,  substituting  the  value  of  tan  Q", 


which  determines  £  -f  J  ^,  whence  ^  and  the  clock  correction.  We 
can  now  find  the  latitude  and  altitude  from  any  one  of  the 
triangles  PSZ,  PS'Z,  PS"Z,  by  NAPIER'S  Analogies  (Sph.  Trig. 
Art.  80)  :  thus,  from  PSZ  we  have 


tan  *  O  +  A)  =  C08  *  (*  +  g)  tan  (45o 
cos  i  («  —  q) 


tan  *  O  -  A)  =  cot  (45 

sin  *  (t  -f  (7) 


(348) 


and  then  <p  =  %  (<p  -f  A)  +  J  (p  —  A),  A  =  J(^  +  h)  -  %(<p  -  h). 

As  all  the  angles  are  determined  by  their  tangents,  an  am- 
biguity exists  as  to  the  semicircle  in  which  they  are  to  be  taken; 
but,  as  GAUSS  remarks,  we  may  choose  arbitrarily  (taking,  for 
example,  §,  §',  Q"  always  less  than  90°,  positive  or  negative 
according  to  the  signs  of  their  tangents),  and  then,  according  to 
the  results,  will  have  in  some  cases  to  make  the  following 
changes  : 

1.  If  the  values  of  <p  and  h  found  by  (348)  are  such  that 
cos^>  and   sin  h  have   opposite   signs,   we   must   substitute 
180°  +  q.  for  q  and  repeat  the  computation  of  these  two  equa- 
tions.    In  this  repetition  the  same  logarithms  will  occur  as 
before,  but  differently  placed. 

2.  If  the  values  of  <p  and  h  exceed  90°,  we  must  take  their 
supplements  to  the  next  multiple  of  180°. 

3.  The  latitude  is  to  be  taken  as  north  or  south  according 
as  sin  <p  and  sin  h  have  the  same  or  different  signs. 

No  ambiguity,  however,  exists  in  practice  as  to  t  -f  J^,  found 
by  (347),  since  Q  —  Q'  can  differ  from  its  true  value  only  by 
180°,  and  this  difference  does  not  change  the  sign  of  cot  (Q  —  Q'}  : 
hence  tan  (t  -f-  £A)  will  come  out  with  its  true  sign;  and  between 


BY   THREE    OR    MORE    EQUAL   ALTITUDES.  289 

the  two  values  of  t  -f  \  I,  differing  by  180°,  or  12*,  the  observer 
will  be  at  no  loss  to  choose,  as  he  cannot  be  uncertain  of  his 
time  by  12*. 

EXAMPLE.  —  Taking  the  example  of  the  preceding  article,  we 
shall  lind 

Q  =  —  37°  57'  9".3     Q'  =  +  6°  17'  51".66     Q"=  —  84°  25'  23".81 

q  =  _  Q  +  Q'  -j-  Q»  =  —  40°  10'  22".85 
t    =  —  39      0  51  .27 

*  (*  +  q)  =  —  39°  35'  37".06         }  (t  —  q)  =  +    0°  34'  45".79 
|  (p  -f  A)  =       52      4  36  .35         *  (>  —  A)  =  —    0    32  44  .84 
=       51    31  51  .5  h    =       52    37  21  .2 


189.  T/"  ?#£  A#tie  observed  more  than  three  stars  at  the  same  altitude, 
we  have  more  than  sufficient  data  for  the  determination  of  the 
latitude  ;  but  by  combining  all  the  observations  we  may  obtain 
a  more  accurate  result  than  from  only  three.  This  combination 
is  effected  by  the  method  of  least  squares,  according  to  which 
we  assume  .approximate  values  of  the  unknown  quantities  and 
then  determine  the  most  probable  corrections  of  these  values,  or 
those  which  best  satisfy  all  the  observations. 

Let  T,  T',  T",  T'",  &c.  be  the  observed  times  by  the  clock 
when  the  several  stars  reach  the  same  altitude.  Let  A  ^be  the 
assumed  clock  correction  at  some  assumed  epoch  ==  T0;  fiTihe 
known  rate.  Let  <p  and  h  be  the  assumed  approximate  values  of 
the  latitude  and  altitude.  With  <p  and  h,  which  will  be  the  same 
for  all  the  stars,  and  with  the  declinations  d,  3',  0",  &c.,  compute 
the  hour  angles  t,  t',  t"  ,  &c.  and  the  azimuths  A,  A',  A",  &c.  If 
the  assumed  values  were  all  correct  and  the  observations  perfect, 
we  should  have  a  +  t  =  T  +  A  T  +  d  T(  T  —  T0)  ;  for  each  of  these 
quantities  then  represents  the  sidereal  time  of  observation  ;  but 
if  ^,  A,  and  AT7  require  the  corrections  dtp,  dh,  and  d&T,  and  if 
dt  is  the  corresponding  correction  of  t,  we  shallthave 


i  +  t-±-<t=  T+  *T  +  d*T 
The  relation  between  dip,  dh,  and  dt  is 

dh  =  —  cos  Ad(p  —  15  cos  <p  sin  Adt 
tu  id  a  similar  equation  of  condition  exists  for  each  star.     In  all 

VOL.  I.—  19 


290  LATITUDE. 

these  equations,  dh  and  dtp  are  the  same,  but  dt  is  different  for 
each.     If  we  put 


/  =  T  +Aar+*r.(r  -  ^0)-(«  +  *) 

/'  r/7/  |          .     /T7       )          ft  ATT    /  /TV  T^  \  /"      f          I        J/  \ 

--  I     +*T+dT(T      -  T0)  —  (a    +O  (349) 

y?^/  rrjn      j       .    /TT     [       *  /TT  x  rrjff  ATT  \  /    "      I      •/•"\ 

&c. 
which  are  all  known  quantities,  we  have 


and  the  equations  of  condition  become 


rfy>  -f-  15  cos  <p  sin  A  .  d&  T-flScos^sin  J.  ./  =0 
rf^  -f  1  5  cos  <f>  sin  A'  .  d  A  7"  -f-  15  cos  <p  sin  A'  ./'  =0 
-f  cosA".d<p  +  IScos^sin^'.^Ar-f-lScos^sinA"./"  —  0 
&c. 


(350) 


from  which,  by  the  method  of  least  squares,  the  most  probable 
values  of  t/A,  dy,  and  d&  T  are  determined.  The  true  values  of 
the  altitude,  latitude,  and  clock  correction  will  then  be  A  -f  dh, 


The  hour  angles  will  be  computed  most  accurately  by  (269), 
which  is  the  same  as  the  following  : 


tan2  *  t  =  8n  *    C  ~  Sin 


cos  I  (C  -f  <f  +  ^)  cos  ^  (C  —  ^>  — 
in  which  f  —  90°  —  A  ;  and  the  azimuths  by 


tan2  }  A  ±=  8n  K^  —  y       ^  cos      C  —  y  — 


cos  J  (C  +  y  -h  *)  sin  *  (C  +  y  —  *) 
Since  ^>  and  £  are  constant,  it  will  be  convenient  to  put 
b  =  J  (C  +  ^  c  ==  i  (C  -  f) 


then 


sin  (c  -4-  J  5)  sin  (b  —  %  ft) 

m  =  -  n  — ^ 1 

cos  (b  -j-  \  5)  cos  (c  —  \  d) 


tan2  Jt  =  mn  tan2  *  A  =  -  (351) 


The  barometer  and  thermometer  should  be  observed  with  each 


BY   THREE    OR    MORE    EQUAL   ALTITUDES.  291 

altitude,  and  if  they  indicate  a  sensible  change  in  the  refraction 
a  correction  for  this  change  must  be  introduced  into  the  equations 
of  condition.  Thus,  if  r0  is  the  refraction  for  the  altitude  h  for 
the  mean  height  of  the  barometer  and  thermometer  during  the 
whole  series,  while  for  one  of  the  stars  it  is  r,  then  the  assumed 
altitude  requires  for  that  star  not  only  the  correction  dh,  but  also 
the  correction  r  —  rQ.  Hence,  if  we  find  the  refractions  r,  r',  r", 
&c.  for  all  the  observations,  and  take  their  mean  r0,  we  have  only 
to  add  to  the  equations  of  condition  respectively  the  quantities 
r  —  r0,  r'  —  r0,  r"  —  r0,  &c. 

If  any  one  of  the  stars  is  observed  at  an  altitude  h^  slightly 
different  from  the  common  altitude  A,  we  correct  the  correspond- 
ing equation  of  condition  by  adding  the  quantity  h  —  hr 

190.  We  may  also  apply  the  preceding  method  to  the  case 
where  there  are  but  three  observations.  The  final  equations  are 
then  nothing  more  than  the  three  equations  of  condition  them- 
selves, from  which  the  unknown  quantities  will  be  found  by 
simple  elimination.  It  will  easily  be  seen  that  this  elimination 
leads  to  the  expressions  for<^>  and  d  A  T  already  given  on  p.  284, 
if  we  there  exchange  dT,  dT' ,  and  dT"  for/,/',  and/"  respect- 
ively. We  can  simplify  the  computation  by  assuming  A  T  so  as 
to  make  one  of  the  quantities  /  /',  /"  zero.  Thus,  we  shall 
have/—  0  if  we  determine  A  J'by  the  formula 


&T=a  ^M  —  [T-f  dT(T—  T0)]  (352) 

then,  finding/7  and/"  with  this  value,  and  putting 

_  sin  £  A'  cos  £  A'  , 

=  sin  \  (A*  —  A)  sin  *  (A"  —  A'} 

sin  M"  cos  &  -A"  n 


~  sin  I  (A"  —  A}  sin  }(A"  —  A') 
we  shall  have  the  following  formulae  : 

d±T=  —  tf  sin  1(4  -f  A")  -f  A"  sin  1(4'  +  A) 
d<f> 


15  cos  <p 


=  _  K  cos  1  (A  +  A")  -f-  k'  cos  i  (A'  +  A) 


— —  =  _|_  ]f  cos  $  (A"  —  A)  —  K'  cos  J  (A'  —  A) 
15  cos  <f> 


(353) 


292  LATITUDE. 

EXAMPLE. — Taking  the  three  observations  above  employed, 
and  assuming  the  approximate  values 

A  T  =  —  11-  0%      ?  =  51°  32'  0",      h  =  52°  37'  0", 
we  shall  find,  by  (351), 

t  =  —  2*  36™  5'.50      f  =  —  &  19-  55'.65      t"  =  3»  23"*58'.25 
A  =  —  66°  15'.2         A'  =  —  177°  50'.2         A"  =  90°  18M 

By  (349),  putting  in  this  case  dT '=  0,  we  then  have 

/  =  —  1-.83        /'  =  -f  80«.95        f"  =  —  6*.21 
and  the  equations  of  condition  (350)  become 

dh  -}-  0.4027  d<p  —  8.5410  d  A  T  +  15.63  =  0 
<ZA  —  0.9993  dy  —  0.3522  d  A  T  —  28.51  =  0 
<ta  _  0,0053  <Z?  -f  9.3308  d  A  T7  —  57.94  =  0 

whence 

d  A  T  =  -f  3'.92        dp  =  —  8".58         <7A  =  +  21".31 

and  the  true  values  of  the  required  quantities  are,  therefore, 
A  T  =  —  10-  56-.08       <p  =  51°  31'  51".42       h  =  52°  37'  21".31 

agreeing  almost  perfectly  with  the  values  before  found. 

Since  in  this  example  there  are  but  three  observations,  we 
may  also  employ  the  formulae  (353),  first  assuming 

A  T  =  —  10™  58M7 
which  is  the  value  given  by  (352).     With  this  we  find 

/'  ==  4-  82'.78  /"  =  —  4'.38 

log  k  =  0.4199  log  A"  =  wO.4932 

and  by  (353)  we  shall  find 

d  A  T  =  +  2«.09        d<?  =  —  8".58        dh  =  +  21".31 

Hence  the  true  clock  correction  is  -  -  10W  58M7  -f-  2*.09  — 
-  10m  56*.08;  and  the  values  of  the  latitude  and  altitude  also* 
agree  with  the  former  values. 


BY    TRANSITS.  293 

191.  We  may  here  observe  that,  theoretically,  the  latitude 
might  be  found  also  from  three  different  altitudes  of  the  same 
star  and  the  differences  of  azimuth  ;  for  we  should  then  have 

sin  d  =  sin  9?  'sin  h    -f  cos  y  cos  h  cos  A 

sin  d  =  sin  y>  sin  h'  -f  cos  <p  cos  h!  cos  (A  -f-  A  ) 

sin  d  =  sin  <p  sin  h"  -j-  cos  <p  cos  h"  cos  (A  -\-  A') 

in  which  A  is  the  azimuth  of  the  star  at  the  first  observation, 
and  the  differences  of  azimuth  /  and  X'  are  supposed  to  be  given. 
The  solution  of  Art.  187  may  be  applied  to  these  equations  by 
writing  h  for  d  and  A  for  t. 

Again,  there  might  be  found  from  three  different  altitudes  of 
the  same  star  not  only  the  latitude  and  time,  but  also  the  decli- 
nation of  the  star  ;  for  we  then  have 

sin  h    =  sin  <p  sin  <5  -j-  cos  <p  cos  d  cos  t 

sin  h'  =  sin  </>  sin  d  -f  cos  <p  cos  d  cos  (t  -f-  >l  ) 

sin  h"  =  sin  <p  sin  d  -j-  cos  <p  cos  d  cos  (£  -{-  A') 

from  which  we  can  readily  deduce  y>,  t,  and  d.  But  the  method 
is  of  no  practical  value,  as  the  errors  of  observation  have  too 
much  influence  upon  the  result. 


NINTH    METHOD.  —  BY   THE    TRANSITS    OF    STARS    OVER    VERTICAL 

CIRCLES. 

192.  "We  may  observe  the  time  of  transit  of  a  star  over  any 
vertical  circle  with  a  transit  instrument  (or  with  an  altitude  and 
azimuth  instrument,  or  common  theodolite)  ;  for  when  the  rota- 
tion axis  is  horizontal,  the  collimation  axis  will,  as  the  instru- 
ment revolves,  describe  the  plane  of  a  vertical  circle.  For  any 
want  of  horizontality  of  the  rotation  axis,  or  other  defects  of 
adjustment,  corrections  must  be  applied  to  the  observed  time  of 
transit  over  the  instrument  to  reduce  it  to  the  time  of  transit 
over  the  assumed  vertical  circle.  These  corrections  will  be 
treated  of  in  their  proper  places  in  Vol.  II.  ;  and  I  shall  here 
assume  that  the  observation  has  been  corrected,  and  gives  the 
clock  time  T  of  transit  over  some  assumed  vertical  circle  the 
azimuth  of  which  is  A.  The  clock  correction  A  T  being  known, 
we  have  the  star's  hour  angle  by  the  formula 

t=  T+ 


LATITUDE. 

and  then,  the  declination  of  the  star  being  given,  we  have  the 
equation  [from  (14)] 

cos  t  sin  <p  —  tan  d  cos  <p  —  sin  t  cot  A 

If,  then,  A  is  also  known,  the  latitude  <p  can  be  found  by  this 
equation.  Let  us  inquire  under  what  conditions  an  accurate 
result  is  to  be  expected  by  this  method.  By  differentiating  the 
equation,  we  find  [see  (51)] 

cos  q  cos  d  tan  £  sin  q 

d<?  =  -  -  dt dA  -\ -  dd 

cos  C  sin  A  sin  A  cos  C  sin  A 

from  which  it  appears  that  sin  A  and  cos  f  must  be  as  great  as 
possible.  The  most  favorable  case  is,  therefore,  that  in  which 
the  assumed  vertical  circle  is  the  prime  vertical,  and  the  star's 
declination  differs  but  little  from  the  latitude  ;  for  we  then  have 
A  =  90°  and  £  small.  Indeed,  these  conditions  not  only  increase 
the  denominator  of  the  coefficient  of  dt,  but  also  diminish  its 
numerator,  since,  by  (10),  we  have 

cos  q  cos  d  =  sin  C  sin  y  -f  cos  C  cos  y  cos  A 

which  vanishes  wholly  when  the  star  passes  through  the  zenith. 
Moreover,  if  the  same  star  is  observed  at  both  its  east  and  west 
transits  over  the  prime  vertical,  we  shall  have  at  one  transit  sin 
A  =  —  1,  at  the  other  sin  A  =  -f-  1,  and  the  mean  of  the  two 
resulting  values  of  the  latitude  will,  therefore,  be  wholly  free 
from  the  effect  of  a  constant  error  in  the  clock  times,  that  is,  of 
an  error  in  the  clock  correction.  It  is  then  necessary  only  that 
the  rate  should  be  known.  This  method,  therefore,  admits  of  a 
high  degree  of  precision,  and  requires  for  its  successful  applica- 
tion only  a  transit  instrument,  of  moderate  dimensions,  and  a 
time-piece.  Its  advantages  were  first  clearly  demonstrated  by 
BESSEL*  in  the  year  1824 ;  but  it  appears  that  very  early  in  the 
last  century  ROMER  had  mounted  a  transit  instrument  in  the 
prime  vertical  for  the  purpose  of  determining  the  declinations  of 
stars  from  their  transits,  the  latitude  being  given.  The  details 
of  this  important  method  will  be  given  in  Vol.  II.,  under 
"  Transit  Instrument." 


*Aatronom.  Nach.,  Vol.  III.  p.  9. 


BY    TRANSITS.  295 

193.  It  may  sometimes  be  possible  to  observe  transits  only  over 
some  vertical  circle  the  azimuth  of  which  is  undetermined.  We 
must  then  observe  either  two  stars,  or  the  same  star  on  opposite 
sides  of  the  meridian.  We  shall  then  have  the  two  equations 

cos  t  .  tan  A  sin  y>  —  tan  3  .  tan  A  cos  y  =  sin  t 
cos  t' .  tan  A  sin  <p  —  tan  d' .  tan  A  cos  <p  =  sin  t' 

from  which  the  two  unknown  quantities  A  and  <p  can  be  deter- 
mined. If  the  same  star  is  observed,  we  shall  only  have  to  put 
df  -=  d.  Regarding  tan  A  sin  <p  and  tan  A  cos  <p  as  the  unknown 
quantities,  we  have,  by  eliminating  them  in  succession, 

sin  t  sin  d'  cos  d  —  sin  t'  c'os  d'  sin  8 
tan  A  sin  <p  = 


tan  A  cos  <p  = 


cos  t  sin  d'  cos  d  —  cos  f  cos  d'  sin  d 

—  sin  (tr  —  t)  cos  $'  cos  d 


cos  t  sin  dr  cos  d  —  cos  t'  cos  d'  sin  d 
If  we  introduce  the  auxiliaries  m  and  Jf,  such  that 

m  sin  M  =  sin  (*'  -f  *)  sin  }  (f  —  f)  )  .  „._ 

m  cos  M  =  sin  (6'  —  *)  cos  }  (T  —  t  )  j  ^ 

we  shall  easily  find 

m  sin  [J  (T  +  0  —  ^]  —  8m  t  sin  5'  cos  d  —  sin  f  cos  3'  sin  5 
m  cos  [2  (t'  -{-  f)  —  Jlf  ]  =  cos  t  sin  fl'  cos  <S  —  cos  t'  cos  5'  sin  8 
m  sin  [J  (f  —  £)  —M]  =  —  sin  (f  —  t)  cos  5'  sin  5 

and  hence 

tan  A  sin  <p  =  tan  [£  (T  -f  f)  —  Jf] 

sin  [*  (f  —  n  —  M  1  cot  5 


tan  A  cos  c?  = 

cos  [J  (f  -f  t)  - 


(356) 


which  determine  A  and  <p  by  a  simple  logarithmic  computation. 
The  solution  will  be  still  more  convenient  in  the  following  form  : 


sin  (3'  —  8) 

.  sin  [^  (f  -4-t}  —  Ml 
tan  <p  =  tan  a  - 

sin  [J(f  —  0  — 


sin  (f> 


(357) 


296  LATITUDE. 

If  the  same  star  is  observed  at  each  of  its  transits  over  the 
same  vertical  circle,  we  have  df  =  S,  and  hence  tan  M  =  &>, 
M~  90°,  which  gives 

tan  A  =     _«**('+  Q      (358) 


cos  i  (t  —  t)  sin  <f> 

If  the  same  star  is  observed  twice  on  the  prime  vertical,  we 
must  have  t'  -J-  t  =  0,  since  tan  A  =  oo  ;  and  then, 

tan  3  tan  3 

tan  <p  =  --  =  -  (359) 

cos  £  (t'  —  i)        cos  * 

which  follows  also  from  (354)  when  cot  A  =  0  ;  or,  geometrically, 
from  the  right  triangle  formed  by  the  zenith,  the  pole,  and  the 
star,  as  in  Art.  19. 

If  the  latitude  is  given,  we  can  find  the  time  from  the  transits 
of  two  stars  over  any  (undetermined)  vertical  circle  by  the  second 
equation  of  (357),  which  gives 


sin  [J  (tf  +  t)  —  M]  =       _    8in  [J  (t'—t)  — 
tan  d 

for  the  observation  furnishes  the  elapsed  time,  and  hence  t'  —  1\ 
and  this  equation  determines  J(/'  -f-  1),  and  hence  both  t  and  t'. 

If  the  latitude  and  time  are  given,  we  can  find  the  declination 
of  a  star  observed  twice  on  the  same  vertical  circle,  by  (358). 
When  the  observation  is  made  in  the  prime  vertical,  this  becomes 
one  of  the  most  perfect  methods  of  determining  decimations. 
See  Vol.  II.,  Transit  Instrument  in  the  Prime  Vertical. 

194.  The   following  brief  approximative   methods   of  deter- 
mining the  latitude  may  be  found  useful  in  certain  cases. 

TENTH    METHOD.  —  BY   ALTITUDES    NEAR    THE    MERIDIAN    WHEN    THE 
TIME    IS    NOT    KNOWN. 

195.  (A.)  By  two  altitudes  near  the  meridian  and.  the  chronometer 
times  of  the  observations,  when  the  rate  of  the  chronometer  is  known, 
but  not  its  correction. 

Let 

h,  h'  =  the  true  altitudes, 

T,  T'  =  the  chronometer  times, 


TWO    ALTITUDES    NEAR   THE    MERIDIAN.  297 

then,  t  and  tf  being  the  (unknown)  hour  angles  of  the  observations, 
we  have,  by  (287),  approximately, 

^  =  h  -f-  at* 
hl==hf  -i-  at'* 

in  which  At  is  the  meridian  altitude,  and 

225  sin  1"  cos  <p  cos  d 
a  =  -  •  -  - 
cos  Aj 

The  mean  of  these  equations  is 


and  their  difference  gives 

h  —  h'  =  a(t'—t')  (f  -f  f) 
But  we  have 

T=i(!r  —  T)  =  $(tf  —  f) 

in  which  we  suppose  the  interval  T1  —  Tto  be  corrected  for  the 
rate  of  the  chronometer.     Hence 

f+t  _\(h—W) 


2  ar 

which,  substituted  in  the  above  expression  for  hv  gives 


ar2 


(360) 


According  to  this  formula,  the  mean  of  the  two  altitudes  is 
reduced  to  the  meridian  by  adding  two  corrections  :  1st,  the 
quantity  ar2,  which  is  nothing  more  than  the  common  "reduc- 
tion to  the  meridian"  computed  with  the  half  elapsed  time  as  the 
hour  angle  ;  2d,  the  square  of  one-fourth  the  difference  of  the 
altitudes  divided  by  the  first  correction. 

If  we  employ  the  form  (285)  for  the  reduction,  we  have 


A1  =  J  (h  +  A')  -f  4m  +  5^A 11  (361) 

in  which 

cos  <p  cos  5  2  sin2  £T 

A  — m  — 


cos  ht  sin  1" 

and  m  is  taken  from  Table  V.  or  log  m  from  Table  VI. 


298 


LATITUDE. 


EXAMPLE  1. — From  the  observations  in  the  example  of  Art. 
171,  I  select  the  following,  which  are  very  near  the  meridian. 


Chronometer. 
23*  50-  46'.5 

_0 0    37.5 

T  =          4~  55  .5 

log  m  1.6778 

log  A  _CM)930 

\ogAm  1.7708 

log  [i(A  —A')]2  2.8198 

log  2d  corr.  1.0490 


Obsd.  alts.  Q 

True  alts.  Q 

50° 

5'  42". 

8     A'   = 

50° 

21' 

1" 

.6 

50 

7  25  . 

5     A    = 

50 

22 

50 

.4 

KA 

-».= 

25 

.7 

*(A 

+  A')  - 

50 

21 

59 

.0 

J.m  = 

_{- 

59 

.0 

2d  corr.  = 

± 

11 

.2 

* 

*,  = 

50 

23 

9 

.2 

:,  = 

39 

36 

50 

.8 

«,  = 

—  1 

48 

9 

,2 

=5      37    48  41  .6 


EXAMPLE  2. — In  the  same  example,  the  first  and  last  observa- 
tions, which  are  quite  remote  from  the  meridian,  are  as  follows : 


Obsd.  alts.  Q 

49°  51'  19^3 
49    50  24 

I  (A 


True  alts.  Q 
h     =  50°  6'  43".7 
h'   =50    5  48  .4 

h')  =  ~         13  .8 


Chronometer. 

23*  37™  35' 
0   18    31 

r  =          20    28 


which  give  Am  =  16'  58 ",  and  the  2d  corr.  —  0".2,  whence 
tp  =  37°  48'  37". 

This  simple  approximative  method  may  frequently  be  useful 
to  the  traveller,  and  especially  at  sea,  where  the  meridian  obser- 
vation has  been  lost  in  consequence  of  flying  clouds.  At  sea, 
however,  the  computation  need  not  be  carried  out  so  minutely 
as  the  above,  and  the  method  becomes  even  more  simple.  See 
Art.  204. 

M.  Y.  CAILLET*  gives  a  method  for  the  same  purpose,  which  is 
readily  deduced  from  the  above.  Put 


Jc  =  h'  —  h 


T,  = 


then  (360)  becomes 


ft   =  A  _j_  _  _f_ 

2  4 

,    (*-f^ 


4ar" 


*  Traite  de  Navigation  (2d  edition,  Paris,  1857),  p.  319. 


THREE    ALTITUDES    NEAR    THE    MERIDIAN.  299 

or,  putting 


2  sin2  }  r' 
m  =  -  Am  = 

sin  V 


(362) 


in  which  /i  is  the  altitude  farthest  from  the  meridian.  Although 
this  reduces  the  two  corrections  of  (361)  to  a  single  one,  the 
computation  is  not  quite  so  simple. 

196.  (B.)  By  three  altitudes  near  the  meridian  and  the  chronometer 
times  of  the  observations,  when  neither  the  correction  nor  the  rate  of  the 
chronometer  is  known.  —  In  this  case  we  assume  only  that  the  chro- 
nometer goes  uniformly  during  the  time  occupied  by  the  observa-. 
tions.  Let 

h,   h',   h"    =  the  true  altitudes, 

T,  T',  T"  =  the  chronometer  times, 

Tl    =  the  chronometer  time  of  the  greatest  altitude. 

If  we  introduce  the  factor  for  rate  =  k,  according  to  Art.  171, 
the  formula  for  the  reduction  to  the  meridian  by  GAUSS'S  method 
is,  approximately, 

ht  =  h  -f  akt2 

in  which  t  is  the  time  reckoned  from  the  greatest  altitude.  De- 
noting ak  by  a,  we  have  then,  from  the  three  observations, 

h^h  +  ^T     -   T,)'  ^| 

ft1  =  A'+a(!T  --  Ttf  }    (363) 

hl  =  h"+»(T"-  TJ*  ) 

which  three  equations  suffice  to  determine  the  three  unknown 
quantities  a,  Tv  and  hr  By  subtracting  the  second  from  the 
first,  and  the  third  from  the  second,  we  obtain 


T  —  T 
h'      -h 


rrin m 

and  the  difference  of  these  is 


-  =  a(T"+   T')  —  2a7\ 


"  '*  /  rrtn 

:r=  a  (  J.       

rrii! /T7'  rnt  rn  N 


300  LATITUDE. 

If,  then,  we  put 

b  =  — =  the  mean  change  of  altitude  in  one  second 

of  the  chronometer  from  the  first  to  the 
second  observation, 

c  =  — —  —  ditto  from  the  second  to  the  third  obser- 
vation, 
we  have 

c  —  b 


rpii  m 

=  T  +  T'        b         YT=T'+T" 

1  o  1 


(364) 


2                   2a                                     2  2a 

Having  thus  found  Tv  we  can  find  \  from  any  one  of  the  equa- 
tions (363),  all  of  which  will  give  the  same  result  if  the  compu- 
tation is  correct.* 

EXAMPLE. — From  the  observations  in  the  example  of  Art.  171 
I  select  the  following  three  observations : 

Obsd.  alts.  Q                    True  alts.  Q  Chronometer. 

50°  5'  42".8            h   =  50°  21'    7".6  T   =  23*  50"  46'.5 

50    7  27  .              h'  =  50    22  51  .9  T'  =  23  55    16  . 

50    7  25  .5            h"  =  50    22  50  .4  T"  =    0     0    37  .5 

h  —  h'  =  —  104".3     T'  —  T  =  269*.5  b  =  —  0.3869 

&'  —  #'  =  +      1.5     T"—  2"  =321 .5  c  =  +  0.0047 

T"—  !T=591.  c  —  b  =       0.3916 


—  —  =  +     4    52  .0  log  (T  —  Ttf  =  5.2604 

2a 

T1=23   57    53.3  log  a  (  T—  Tt)2  =  2.0817 

T  —  Tj  =  —  7*  6*.8  h  =    50°  21'    7".6 

a  (T  —  TJ2  =    -f      2     0  .7 

Aj  =    50    23     8  .3 

C,  =    39    36  51  .7 

t±  =  —  1    48    9  .2 

<p  =    37    48  42  .5 

The  mean  of  the  three  values  found  from  these  altitudes  in  Art. 
172  is  37°  48'  42".8. 

*  This  method  is  essentially  the  same  as  that  proposed  by  LITTROW  (Astronomic, 
Vol.  I.  p.  171.)  I  have  here  rendered  it  applicable  to  the  sun  without  considering 
the  change  of  declination,  by  introducing  GAUSS'S  form  for  the  reduction  to  the 
meridian. 


REDUCTION    TO    MERIDIAN    BY    AZIMUTHS.  301 

197.  (C.)  By  two  altitudes  or  zenith  distances  near  the  meridian 
and  the  difference  of  the  azimuths. — If  the  observer  has  no  chrono- 
meter, he  may  still  obtain  his  latitude  by  circummeridian  alti- 
tudes, if  he  observes  the  altitudes  with  a  universal  instrument, 
and  reads  the  horizontal  circle  at  each  observation,  taking  care, 
of  course,  that  the  star  is  always  observed  at  the  middle  vertical 
thread.  As  this  instrument  generally  gives  directly  the  zenith 
distances,  we  shall  substitute  £  for  90°  —  h.  "We  have  the  equa- 
tion 

sin  S  =  sin  <p  cos  C  —  cos  <p  sin  £  cos  A 

=  sin  (<p  —  C)  +  2  cos  <p  sin  C  sin2  J  A 
whence 

cos  I  (^  -f  d  —  C)  sin  i  [C  —  (<f>  —  <5)]  =  cos  p  sin  C  sin2  £  A 
But 

<p  —  d  =  Ct  =  the  meridian  zenith  distance; 

and  hence 

8in  i  (C  _  Ci)  =  cosy  Bin  CBin'M  (365 

' 


which  expresses  the  reduction  to  the  meridian  =  f  —  £\  when 
the  absolute  azimuth  A  is  given.  If  the  observation  is  very 
near  the  meridian,  we  may  neglect  J  (f  —  fj  in  the  denominator 
of  the  second  member,  and  take 

,cos  v  sin  C«     2  sin2  £  A 

C  —  Ct  — 


cos  d  sin  1" 
or,  putting 

^cOMPBtaC,  sinT 

cos   5  2 

C  —  C,  =  aA*  (367) 

from  which  it  follows  that  near  the  meridian  the  zenith  distance 
varies  as  the  square  of  the  azimuth. 

Now,  when  we  have  taken  two  observations,  we  have 


whence,  putting 


302  LATITUDE. 

we  deduce  the  following  equation,  analogous  to  (360), 


(368) 


Here  r  is  equal  to  one-half  the  difference  of  the  readings  of  the 
horizontal  circle,  and  is  therefore  known  ;  and  the  computation 
is  entirely  similar  to  that  of  the  formula  (360). 

198.  (D.)  By  three  altitudes  or  zenith  distances  near  the  meridian 
and  the  differences  of  azimuths. 

Supposing  the  observations  taken  with  a  universal  instru- 
ment, let 

C,  C',  C"  =  the  true  zenith  distances, 
A,  A'y  A"  —  the  readings  of  the  horizontal  circle, 

we  shall  have,  by  the  preceding  article, 

Cj=C   —a,  (A   —A^  ^ 

Ct  =  C'  —  a  (A'  —  Ay  I    (369) 

in  which  Al  is  the  (unknown)  circle  reading  in  the  meridian, 
and  a  is  the  (unknown)  change  of  zenith  distance  for  V  of  azi- 
muth. These  equations  are  solved  in  the  same  manner  as  (363) ; 
and  hence  we  have  the  formulae 


c  -~ 


A'  — A  A"  —  A' 

c  —  b 
~-~J^A 
A  +  A        b  _A'+A"       c 

jjj    ,/jL      —   — —   


(370) 


which  determine  a  and  Av  after  which  fjs  found  by  any  one  of 
the  equations  (369).* 


*  In  this  connection,  see  an  article  by  LITTROW  in  ZACH'S  Monatliche  Correspondenz, 
Vol.  X.  (1824). 


BY  CHANGE  OF  ALTITUDE.  303 

ELEVENTH  METHOD. — BY  THE  RATE  OF  CHANGE  OF  ALTITUDES  NEAR 
THE  PRIME  VERTICAL.* 

199.  We  have,  Art  149, 

—  =  cos  <f>  sin  A 

15  dt 

If  then  we  observe  two  altitudes  near  the  prime  vertical  in  quick 
succession,  noting  the  times  by  a  stop-watch  with  as  great  pre- 
cision as  possible,  and  denote  the  difference  of  the  altitudes,  or 
of  the  zenith  distances,  by  d£,  and  the  difference  of  the  times  by 
dt,  we  shall  have 

cos  <p  =  — —  cosec  A  (371) 

15  dt 

The  observation  being  made  near  the  prime  vertical,  an  error  in 
the  supposed  azimuth  A  will  have  but  small  influence  upon  the 
result.  If  the  observation  is  exactly  in  the  prime  vertical,  or 
within  a  few  minutes  of  it,  we  may  put 

cos  <f>  =  — —  (372) 

15  dt 

This  exceedingly  simple  method,  though  not  susceptible  of 
great  precision,  may  be  very  useful  to  the  navigator,  as  it  is 
available  when  the  sun  is  exactly  east  or  west,  and,  consequently, 
when  no  other  method  is  practicable,  and,  moreover,  requires 
no  previous  knowledge  of  the  time  or  the  approximate  latitude, 
or  of  the  star's  decimation. f 

EXAMPLE. — 1853  July  3,  PRESTEL  observed,  near  the  prime 
vertical,  the  time  required  by  the  sun  to  change  its  altitude  by  a 
quantity  equal  to  its  apparent  diameter,  by  observing  with  a 
sextant  first  the  contact  of  the  lower  limb  with  its  image  in  an 
artificial  horizon,  and  then  the  contact  of  the  upper  limb  with 


*  PRESTEL,  in  Astron.  Nach.,  Vol.  XXXVII.  p.  281. 

f  Since  the  star's  declination  is  not  required,  this  method  has  the  additional 
advantage  (which  may  at  times  be  of  great  importance  to  the  traveller)  of  being 
practicable  ivithout  the  use  of  the  Ephcmeri*.  This  feature  entitles  this  method  to  a 
prominent  place  in  works  on  navigation. 


304  LATITUDE   AT    SEA. 

its  image,  the  sextant  reading  being  the  same  at  both  observa- 
tions, namely,  30°  15'  0".     He  found 

Chronometer. 

Contact  of  lower  limb,   4*  43™  34'.     P.M. 
"  upper     "       4  47      5.5 

3    31.5 

The  sun's  diameter  was  31'  32".     Hence  we  have 

d:  =  31'  32"    ==  1892"                  log  3.2769 

dt  =    3™  31-.5  =    211-.5                ar.  co.  log  7.6747 

log  Jj  8.8239 

<P  =  53°  23'.5                                 log  cos  9  9.7755 

The  azimuth,  however,  was  not  exactly  90°,  but  about  88°  20'. 
Hence  we  shall  have,  more  exactly, 

9.7755 

A  =  88°  20'  log  cosec  A  0.0002 

<p  =  53    22.3  log  cos  <p      9.7757 

It  is  evident  that  the  method  will  be  more  precise  in  high  lati- 
tudes than  in  low  ones. 

FINDING   THE    LATITUDE   AT    SEA. 

First  Method. — By  Meridian  Altitudes. 

200.  This  is  the  most  common,  as  well  as  the  simplest  and 
most  reliable,  of  the  methods  used  by  the  navigator.  The  alti- 
tude is  observed  with  the  sextant  (or  quadrant)  from  the  sea 
horizon,  and,  in  addition  to  the  corrections  used  mi  shore,  the 
dip  of  the  horizon  is  to  be  applied.  The  true  altitude  being 
deduced,  the  latitude  is  found  by  (277)  or  (278),  Art.  161. 

At  sea  the  time  is  seldom  so  well  known  as  to  enable  the 
navigator  to  take  the  star  at  the  precise  instant  of  its  meridian 
passage.  But  the  meridian  altitude  of  a  star  is  distinguished  as 
the  greatest,  to  secure  which  the  observer  commences  to  measure 
the  star's  altitude  some  minutes  before  the  approximately  com- 
puted time  of  passage,  and  continues  to  observe  it  until  he  per- 
ceives it  to  be  falling.  The  greatest  of  all  his  measures  is  then 
assumed  as  the  meridian  altitude. 


MERIDIAN    ALTITUDES.  305 

The  most  common  practice  in  the  case  of  the  sun  is  to  bring 
the  lower  limb,  reflected  in  the  mirrors  of  the  instrument,  to 
touch  the  sea  horizon  seen  directly  (a  few  minutes  before  noon), 
and  then  by  the  tangent  screw  to  follow  the  sun  as  long  as  it 
rises,  never  reversing  the  motion  of  the  screw  ;  as  soon  as  the 
sun  begins  to  fall,  the  limb  will  appear  "to  dip"  in  the  sea  by 
lapping  over  the  line  marking  the  horizon.  Hence,  when  the 
sun  "dips,"  the  observation  is  complete,  and  the  instrument  is 
read  off.  But,  as  the  waves  of  the  sea  cause  the  ship  to  rise  and 
fall,  the  depression  of  the  sea  horizon  is  constantly  fluctuating 
by  the  small  amount  due  to  the  change  in  the  height  of  the 
observer's  eye  :  it  is,  consequently,  impossible  to  keep  the  sun's 
reflected  image  in  constant  contact  with  the  horizon.  Expe- 
rienced observers  advise,  therefore,  to  observe  and  read  off 
separate  altitudes  in  rapid  succession,  continuing  until  the 
numbers  read  off  decidedly  decrease  ;  the  greatest  is  then  taken 
as  the  meridian  altitude,*  or,  still  more  accurately,  the  mean  of 
the  greatest  and  the  two  immediately  adjacent  may  be  taken  as 
the  meridian  altitude,  free  from  the  inequalities  produced  by  the 
motion  of  the  eye. 

201.  The  greatest  altitude,  however,  is  not  the  meridian  alti- 
tude, except  in  the  case  of  a  fixed  star.  To  find  the  correction 
for  a  change  of  declination,  we  have,  for  the  time  (#)  from  noon 
when  the  sun  is  at  the  greatest  altitude,  the  formula  (294),  or 

A<5  sin    c0  —  <S 

~ 


810000  sin  I"     cos  <p  cos<5 

in  w^hich  A£  is  the  hourly  change  of  declination  expressed  in 
seconds.  The  reduction  of  the  maximum  altitude  to  the  meri- 
dian altitude  is  the  quantity  ?/,  Art.  172,  or 

(15  »5)2  sin  1"    cos 


2  sin  O  —  <5) 

These  formulae  give  $  in  seconds  of  time  and  y  in  seconds  of  arc. 
For  nautical  use,  let 

a  =  the  change  of  altitude  (expressed  in  seconds  of  arc)  in- 
one  minute  of  time  from  the  meridian; 

*  RAPER,  Practice  of  Navigation  (4th  edition,  1852),  p.  226. 
VOL.  I.—  20 


30G  LATITUDE   AT    SEA. 

then,  by  (287),  putting  t  =  60', 

_  810000  sin  I"    cos  y  cos  d 

~~2~~      ' '  sin  O  —  <5) 
and  therefore 


__/^\2  J_ 

\~60/  4a 


The  value  of  a  is  given  in  BOWDITCH'S  Navigator,  Table  XXXII., 
with  the  arguments  <p  and  a. 

If  ice  express  AO  in  minutes  of  arc,  we  shall  have  #  in  minutes  of 
time  and  y  in  seconds  of  arc,  by  the  formulae* 

-         - 


These  formulae  may  be  used  also  for  the  moon  or  a  planet.  The 
greatest  value  of  A^  for  the  sun  is  1',  namely,  at  the  equinoxes 
when  e>  =  0;  and  in  this  case,  if  the  latitude  is  70°,  we  have 
a  =  0.7  and 

1" 


4X0.7 


=  0".36 


a  quantity  altogether  insensible  in  nautical  practice. 

For  the  moon,  however,  we  may  have  A£  =  18',  and  for 
<p  =  70°  the  least  value  of  a  =  0.6,  whence 

=     (18)2     ^135"  =  2'  is// 
4X0.6 

Even  this  (which,  it  must  be  observed,  is  for  an  extreme  case) 
is  usually  neglected  by  navigators,  who  regard  observations  of 
the  moon  for  latitude  as  but  approximations,  on  account  of  the 
frequent  indeterminate  character  of  the  sea  horizon  as  seen 
under  the  moon.f 

202.  When  the  ship  is  in  motion,  the  change  of  latitude  pro- 
duces the  same  effect  upon  the  observed  maximum  altitude  as 
an  equal  change  of  declination.  Thus,  as  in  the  last  example 
of  the  preceding  article,  if  a  ship  in  latitude  70°  sail  due  north 

*  BOWDITCH,  Practical  Navigator,  p.  169. 

t  RAPER,  Practice  of  Navigation  (4th  edition),  pp.  177,  226,  230. 


REDUCTION   TO    THE    MERIDIAN.  307 

or  due  south  at  the  rate  of  18  miles  per  hour,  the  maximurrr. 
altitude  will  exceed  the  meridian  altitude  b    2'  15". 


Second  Method.  —  By  Reduction  to  the  Meridian  when  the  Time  is 

given. 

203.  When  the  meridian  observation  is  lost  in  consequence 
of  clouds,  circummeridian  altitudes  may  sometimes  be  obtained. 
The  most  convenient  method  of  reducing  them  at  sea  is  that  of 
BOWDITCH.  In  his  Table  XXXII.  he  gives  the  value  of  a  com- 
puted by  (373)  ;  and  in  Table  XXXIII.  the  value  of  t\  t  being 
reduced  to  minutes.  Each  observed  altitude  h  is  then  reduced 
to  the  meridian  altitude  hv  by  the  formula  (287),  or 

^  =  h  -f  aZ2  (375) 

and  a  number  of  altitudes  are  reduced  at  once  by  the  same 
formula,  by  taking  for  h  the  mean  of  all  the  altitudes,  and  for  t2 
the  mean  of  all  the  values  of  t2.  If  the  observer  has  no  tables, 
he  can  readily  compute  a  by  the  formula 


a  =  1".9635  ^L^rsr  —  [0.2930]  ™  (376) 

s\n(<f>  —  8)       L  Jsm(^  —  <5) 

BOWDITCH'S  table  for  t2  extends,  however,  only  to  t  =  13™. 
When  the  observations  are  more  than  13'"  from  the  meridian, 
he  reduces  the  observation  to  the  meridian  by  the  formula  (282), 

cos  d  =  sin  h  -f-  cos  <p  cos  8  (2  sin2  ?  £) 

employing  a  table  of  log.  versed  sines  for  the  value  of  2  sin2  J^; 
a  table  of  natural  sines  for  sin  h  and  cos  £t ;  and  the  table  of 
logarithms  of  numbers  for  the  value  of  the  last  term.  I  prefer 
the  formula  (283), 

cos  y  cos  3  sin2 if 
sin  J  (At  —  A)  =  — 

which  effects  the  reduction  by  a  single  table. 

Third  Method. — By  Tivo  Altitudes  near  the  Meridian  when  the  Time 

is  not  known. 

204.  As  it  frequently  happens  at  sea  that  the  local  time  is 
uncertain,  the  method  I  have  proposed  in  Art.  195  will  be  found 


308  LATITUDE   AT    SEA. 

of  great  use  to  the  navigator.  Any  two  altitudes  h  and  hr  being 
observed  near  the  meridian,  r  being  one-half  the  chronometer 
interval  between  them,  corrected  for  rate,  expressed  in  minutes, 
and  a  being  fonnd  by  (376),  or  from  BOWDITCH'S  Table  XXXII. , 
we  have  the  meridian  altitude  by  the  formula 


which  may  be  computed  without  the  use  of  logarithms. 

EXAMPIE. — The  approximate  latitude  being  38°  K,  the  de- 
clination at  noon  1°  48'  9"  S.,  the  height  of  the  eye  above  the 
sea  19  feet,  suppose  the  following  observations  taken : 

Chronometer.  Q 

T  =  8*    0-  22'.5  h  =  50°~il'  40" 

Tf  =  Sh  10™  13'.5  -  h'  =  50    10     0 

2)_9_51_0  h  —  h'  =  ~       1  40 

r2    =       24.2  i  (h  +  /O  =  50    10   50 

a    =        2" A         ar*  =  1st  corr.  =  58 

[j  (h  —  A')p  =          625        «£*  =--  2d      "     =  11 

Merid.  alt.  Q  =  50  11  59 

Dip  4  16 

Semidiameter  =  -f-  16  6 

Refr.  and  par.  =  -  42 

hi  =  50    28      7 
Ct  =  39   36    53  1ST. 

^  =    I    48 9  S. 

?  =  37~48~~44  N. 

The  accuracy  of  the  result  depends  in  a  great  degree  upon 
the  accuracy  with  which  the  difference  of  altitude  is  obtained. 
If  in  the  above  example  this  difference  had  been  2r  40",  or  V 
too  great,  we  should  have  found  J(A — A')  =  40",  and  the  2d 
correction  =  ifjp  =  28"  :  consequently  the  resulting  latitude 
would  have  been  only  17"  too  small.  Since  the  same  causes  of 
error,  such  as  displacement  of  the  sea  horizon  by  extraordinary 
refraction,  unknown  instrumental  errors,  &c.,  affect  both  altitudes 
alike,  the  difference  will  usually  be  obtained,  even  at  sea,  within 
a  quantity  much  less  than  V.  The  most  favorable  case  is  that 


THREE    ALTITUDES    NEAR   THE    MERIDIAN.  S09 

in  which  the  altitudes  are  equal  and  the  2d  correction,  conse- 
quently, zero.  It  will  be  well,  therefore,  always  to  endeavor  to 
obtain  altitudes  on  opposite  sides  of  the  meridian. 

We  may  also  obtain  the  time,  approximately,  from  the  same 
observation  ;  for  the  mean  of  the  hour  angles  is,  Art.  195, 


which  is  the  apparent  time  from  noon  at  the  middle  instant 
between  the  observations,  (in  minutes,  r  being  in  minutes,  h  —  hf 
and  a  being  in  seconds)  ;  and  this  time  will  be  before  or  after  noon 
according  as  the  second  altitude  is  greater  or  less  than  the  first. 
Thus,  in  our  example,  we  have 


25 


ar  2.4  X  4.9 

or  the  apparent  time  at  the  middle  instant  was  2m  6s  after  noon. 
The  first  observation  was,  therefore,  2™  49*  before  noon,  and  the 
second  lm  1*  after  noon. 


Fourth  Method.  —  By  Three  Altitudes  near  the  Meridian  when  the 
Time  is  not  known. 

205.  The  method  of  Art.  196  does  not  require  even  the  rate 
of  the  chronometer  to  be  known  ;  but  it  is  hardly  simple  enough 
for  a  common  nautical  method.  But  a  very  simple  method  will 
be  obtained  if  we  take  three  altitudes  at  equal  intervals  of  time. 
Suppose  the  second  altitude  is  observed  at  the  (unknown)  time 
J'from  the  meridian  passage,  the  first  at  the  time  T  —  z,  the 
third  at  the  time  T  -\-  x;  then  we  have,  by  (363), 

hl=h   +  a(T—  xy 
hl=hr  +  a  T2 


Subtracting  the  half  sum  of  the  first  and  third  equations  fro 
the  second,  we  deduce 


310  LATITUDE    AT    SEA. 

The  difference  of  the  first  and  third  gives 


,T  = 


x 


which  substituted  in  the  second  equation, gives  hr 

If  then  we  put  a  for  ax2,  the  computation  is  expressed  by  the 
following  simple  formulae : 


a  =  A'  —  *  (A  -f  A") 


(377) 


EXAMPLE. — The  following  three  altitudes  were  observed  at 
equal  intervals  of  time  near  the  meridian  : 

h  =  43°  8'  20"               V  =  43°  15'  30"  h"  =  43°  4'  0" 

J(fc-|-ft")=43 6  10 

«.  =  9  20  =  560" 

\  (h  —  A")  =  1     5  =    65 

Hence  the  reduction  of  the  middle  altitude  to  the  meridian  is 

[j(A  -  7Q]«  =  65*  _=8;, 
a  560 

which  added  to  A'  gives 

At  =  43°  15'  38" 

Instead  of  equal  intervals  of  time,  we  may  employ  equal  inter- 
vals of  azimuth  (Art,  197),  and  still  reduce  the  altitudes  by  (377); 
but  this  would  be  practicable  only  on  land. 

Fifth  Method. — By  a  Single  Altitude  at  a  given  Time. 

206^  This  is  the  method  of  Art.  164,  which,  however,  should 
be  restricted,  at  sea,  to  altitudes  taken  not  more  than  one  hour 
from  the  meridian,  as  the  time  is  always  imperfectly  known  and 


ALTITUDE    NEAR    THE    PRIME    VERTICAL.  311 

the  error  in  the  latitude  produced  by  an  error  in  the  time 
increases  very  rapidly  as  the  star  leaves  the  meridian  and  ap- 
proaches the  prime  vertical  (Art.  166),  and  the  method  fails 
altogether  when  the  star  is  in  the  prime  vertical.  It  may,  how- 
ever, sometimes  be  very  important  to  determine  the  latitude,  at 
least  approximately,  when  the  sun  is  nearly  east  or  west;  and 
then  the  following  method  may  be  used. 

Sixth  Method. — By  the  change  of  Altitude  near  the  Prime  Vertical. 

207.  This  is  the  method  of  Art.  199.     In  the  morning,  when 
the  sun  has  arrived  within  1°  of  the  prime  vertical  as  observed 
with  the  ship's  compass,  bring  the  image  of  the  sun's  upper 
limb,  reflected  by  the  sextant  mirrors,  into  contact  with  the  sea 
horizon,  and  note  the  time ;  let  the  sextant  reading  remain  un- 
changed, and  note  the  time  when  the  contact  of  the  lower  limb 
occurs.     In  the  afternoon,  begin  with  the  lower  limb.     Then, 
taking  the  sun's  semidiameter  —  S  from  the  almanac,  and  put- 
ting the  difference  of  the  chronometer  times  =  r,  we  have 

cos  y  =  —  =  [9.1249]  -  (378) 

15r 

This  is  evidently  but  a  rough  method,  only  to  be  resorted  to  in 
cases  of  emergency.  With  the  greatest  care  in  observing  the 
contacts,  and  in  latitudes  not  less  than  45°,  the  result  cannot  be 
depended  upon  within  from  five  to  ten  minutes ;  but  even  this 
degree  of  accuracy  may,  in  many  cases  at  sea,  be  quite  satis- 
factory. 

Seventh  Method. — By  the  Pole  Star. 

208.  This  method,  though  confined  in  its  application  to  north 
latitudes,  is  very  useful  at  sea,  as  it  is  available  at  all  times  when 
the  star  is  visible  and  the  horizon  sufficiently  distinct,  and  does 
not   require   a  more   accurate  knowledge  of  the  time  than  is 
usually  possessed  on  shipboard.     The  complete  discussion  of  it 
has  been  given  in  Art.  176 ;  but  for  those  who  wish  only  the 
nautical  method,  and  have  passed  over  that  article,  I  add  the 
following  simple  investigation,  which  is  sufficiently  precise  for 
the  purpose. 

Let  ZN,  Fig.  27,  be  the  meridian ;  Z  the  zenith  of  the  ob- 
server ;  P  the  pole  ;  AN  the  horizon  ;  S  the  star,  which  describes 


312  LATITUDE    AT    SEA. 

a  small  circle  ST  about  the  pole  at  the  dis- 
tance PS  =  p;  ZSA  the  vertical  circle  of  the 
star  at  the  time  of  the  observation  ;  £4  the 
true  altitude  =  A,  deduced  from  the  observed ; 
SPZ  the  star's  hour  angle  =  t ;  PN  the  lati- 
tude =  (p. 

Draw   SB  perpendicular    to    the    meridian : 
then,  since  SP  is  small  in  the  case  of  the  pole 
star  (about  1°  30'),  we  may  regard  PSJ3  as  a 
plane  triangle,  and  hence  we  have 

PB  =  PS.  cos  SPB  =  pcost 
and,  since  BN  differs  very  little  from  SA, 

PN  =  BN  —  PB  =  SA  —  PB 

that  is,* 

</>  =  h  —  p  cos  t 

If  we  put 

0  —  the  sidereal  time, 

a  —  the  star's  right  ascension, 

we  have 

t  =  O  —  a 

and  hence 

(f  =  h  —  p  cos  (0  —  a)  (379) 

If  then  p  and  a  be  regarded  as  constant,  the  term  p  cos  (0  — a) 
may  be  given  in  a  table  with  the  argument  0,  as  in  BOWDITCH'S 
Navigator,  p.  206.  But  the  polar  distance  and  right  ascension 
of  the  pole  star  vary  so  rapidly  that  in  a  few  years  such  a  table 
affords  but  a  rude  approximation.  The  direct  computation  of 
the  formula  with  the  values  of  p  and  a  obtained  from  the 
Ephemeris  for  the  day  of  the  observation  is  preferable. 

EXAMPLE. — 1856  March  10,  from -an  altitude  of  Polaris  ob- 
served from  the  sea  horizon,  the  true  altitude  h  was  deduced  as 
below.  The  time  was  noted  by  a  Greenwich  chronometer 
which  was  fast  5"  30s.  The  longitude  was  150°  0'  W. 

*  If  we  compare  this  with  the  more  exact  formula  (300),  we  see  that  the  error  of 
the  nautical  method  is  \  p1  sin  1"  sin2  Han  A,  which  is  a  maximum  for  t  =  90°. 
Taking  p  =  1°  30',  this  maximum  is  70".7  tan  ^>,  which  amounts  to  3'  when  0  = 
68°  30'. 


BY    TWO    ALTITUDES.  813 

Chronometer         19*  12"  42'  h  ±tt  31°  10'. 

Correction  5    30 

Gr.  M.  T.  ~T9     7    12         p  =  1°  27'  18" 

Longitude  10     0      0  =  87'.3 

Local  M.  T.         ~9     7    12   logj?  1.9410 

Sid.  T.  Gr.  noon    23    13    23   log  cost      n9.5234 

Corr.  for  19A  7m     +38  log  p  cos  t  nl.464$  —  pcost=+    29.1 

6  =      8~23~43  <p  =  31    39 .1 

a  =      1     5    44 

t  =      7   17    59 
=  109°  29'   45" 

Eighth  Method. — By  Two  Altitudes  with  the  elapsed  Time  between 

them. 

209.  This  method  may  be  successfully  applied  at  sea,  and  is 
the  most  reliable  of  all  methods,  next  to  that  of  meridian  or  cir- 
cummeridian  altitudes.  The  formuhe  fully  discussed  in  Arts. 
178  to  183  may  be  directly  applied  when  the  position  of  the  ship 
has  not  changed  between  the  observations. 

But,  since  there  should  be  a  considerable  difference  of  azimuth 
between  the  observations,  the  change  of  the  ship's  position  in 
the  interval  will  generally  be  sufficiently  great  to  require  notice. 
All  that  is  necessary  is  to  apply  a  correction  to  the  altitude  ob- 
served at  the  first  position  of  the  ship,  to  reduce  it  to  what  it  would 
have  been  if  observed  at  the  second  position  at  the  same  instant. 
To  obtain  this  correction,  let  Z',  Fig.  28,  be 
the  zenith  of  the  observer  at  the  first  observa- 
tion, S  the  star  at  that  time ;  Z  his  zenith  at 
the  second  observation,  and  S'  the  star  at  that 
time.  The  first  observation  gives  the  zenith 
distance  Z'S,  the  second  the  zenith  distance 
ZS'.  Joining  the  points  S  and  S'  with  the 
pole  P,  it  is  evident  that  the  hour  angle  SPS' 
is  obtained  from  the  observed  difference  of 
the  times  of  observation  precisely  as  if  the 
observer  had  been  at  rest.  We  have,  there- 
fore, only  to  find  ZS  in  order  to  have  all  the  data  necessary  for 
computing  the  latitude  of  Z  by  the  general  methods. 

The  number  of  nautical  miles  run  by  the  ship  is  the  number 
of  minutes  in  the  arc  ZZf ;  and,  since  this  will  always  be  a  suffi- 


314  LATITUDE    AT    SEA. 

ciently  small  number,  if  we  draw  ZA  perpendicular  to  SZ',  we 
may  regard  ZAZ'  as  a  plane  triangle,  and  take 

ZS  =  Z'S  —  AZ' 
or 

ZS  =  Z'S  —  ZZ'  cos  ZZ'S  (380) 

The  angle  ZZ'S  is  the  difference  between  the  azimuth  of  the 
star  at  the  first  observation  and  the  course  of  the  ship;  and  this 
azimuth  is  obtained  with  sufficient  accuracy  by  the  compass.* 

Employing  the  zenith  distance  thus  reduced  and  the  other 
data  as  observed,  the  latitude  computed  by  the  general  method 
will  be  that  of  the  second  place  of  observation.  In  the  same 
manner  we  can  reduce  the  second  zenith  distance  to  the  place  of 
the  first,  and  then  the  latitude  of  the  first  place  will  be  found. 

210.  The  problem  of  finding  the  latitude  from  two  altitudes  is 
most  frequently  applied  at  sea  in  the  case  where  the  sun  is  the 
observed  body,  the  observation  of  the  meridian  altitude  having 
been  lost.  The  computation  is  then  best  carried  out  by  the 
formula  (315),  (316),  (317),  (318),  employing  for  8  the  mean 
declination  of  the  sun, — i.e.  the  declination  at  the  middle  time 
between  the  two  observations, — and  then  applying  to  the  result- 
ing latitude  the  correction  A^  found  by  the  formula  (323).  To 
save  the  navigator  all  consideration  of  the  algebraic  signs  in 
computing  this  correction,  it  will  be  sufficient  to  observe  the 
following  rule :  1st.  When  the  second  altitude  is  the  greater,  apply 
this  correction  to  the  computed  latitude  as  a  northing  when  the 
sun  is  moving  towards  the  north,  and  as  a  southing  when  the  sun 
is  moving  towards  the  south ;  2d.  When  the  first  altitude  is  the 
greater,  apply  the  correction  as  a  southing  when  the  sun  is  moving 
towards  the  north,  and  as  a  northing  when  the  sun  is  moving 
towards  the  south. 

*  If  we  wish  a  more  rigorous  process,  we  must  consider  the  spherical  triangle 
ZZ'S,  in  which  we  have  the  observed  zenith  distance  Z'S  =  (£'),  the  required  zenith 
distance  ZS  =  C,  the  distance  run  by  the  ship  Z'Z  =  d,  the  difference  of  the  star's 
azimuth  and  the  ship's  course  ZZ'S'  =  a,  and  hence 

cos  C  =  cos  f '  cos  d  -f  sin  £'  sin  d  cos  a 
which  developed  gives 

£  =  £'  —  d  cos  a  -f  %  &  sin  1"  co*  C'  s'in2  a 
the  last  term  of  which  expresses  the  error  of  the  formula  given  in  the  text. 


BY   TWO    ALTITUDES. 

If  the  computer  chooses  to  neglect  this  correction,  he  should 
employ  the  mean  declination  only  when  the  middle  time  is 
nearer  to  noon  than  the  time  of  the  greater  altitude.  In  all  other 
cases  he  should  employ  the  decimation  for  the  time  of  the 
greater  altitude  (Art.  183). 

211.  DOUWES'S  method  of  "double  altitudes."*  —  This  is  a  brief 
method  of  computing  the  latitude  from  two  altitudes  of  the  sun, 
which,  though  not  always  accurate,  is  yet  sufficiently  so  when 
the  interval  between  the  observations  is  not  more  than  1*,  and 
one  of  them  is  less  than  V1  from  the  meridian. 

Let  h  and  hf  be  the  true  altitudes,  3  the  declination  at  the 
middle  time,  jTand  T'  the  chronometer  times  of  the  observa- 
tions, t  and  tr  the  hour  angles.  The  elapsed  apparent  time  X  is 
found  from  the  times  Tand  T'  by  (322),  but  it  is  usually  suffi- 
cient to  take  l=T'—T.  "We  then  have  t'  =  t  +  X  ;  and  by  the 
first  of  (14)  we  have 

sin  h  =  sin  y  sin  3  -j-  cos  y  cos  3  cos  t 

sin  h'  =  sin  <p  sin  d  -j-  cos  y  cos  d  cos  (t  -j-  A) 

The  difference  of  these  equations  gives 

sin  h  —  sin  hf  =  2  cos  <p  cos  d  sin  (t  -j-  2  ^)  sin  J  A 
If  we  put  tQ  —  the  middle  time,  or 


0 
we  deduce 

sin  h  —  sin  h' 

2  sin  t0=  —  (381) 

cos  <p  cos  3  sin  J  A 

which  gives  t0  by  employing  the  supposed  latitude  for  <p  in  the 
second  member.     We  then  have 


and  the  meridian  zenith  distance  f1  is  found  from  the  greater 
altitude  h  by  the  formula  (Art.  168) 

cos  Cj  —  sin  h  -j-  cos  <p  cos  d  (2  sin2  J  #) 

*  The  method  of  finding  the  latitude  by  two  altitudes  is  commonly  called  by  navi- 
gators "the  method  of  double  altitudes,"  —  an  obvious  misnomer,  as  double  means 
twice  the  same. 


316  LATITUDE    AT    SEA. 

and  finally  the  latitude  by  the  formula  <p  —  £\  -f  d.  Since  we 
employ  an  assumed  approximate  latitude,  we  shall  have  to  repeat 
the  process  when  the  computed  latitude  differs  much  from  the 
assumed. 

This  is  the  form  of  the  method  as  proposed  by  DOUWES  and 
adopted  in  BOWDITCH'S  Navigator;  but  the  following  form  is  still 
more  simple,  as  it  requires  only  the  table  of  logarithmic  sines. 
The  formula  for  /0  may  be  written  thus  : 


__  cos  i  (h  -f  hr)  sin  £  (h  —  h'} 
tQ  —  - 

then,  as  before, 


sin 

cos  <  cos  d  sin 


and  the  reduction  of  h  to  the  meridian  altitude  hl  is  found  by 

(283), 

cos  c?  cos  fJ  sin2  \t 

sm  \  (h,  —  h)  =  - 

cos  J  (7^+  /i) 

Adding  At  —  h  to  A,  we  have  the  meridian  altitude,  from  which 
the  latitude  is  deduced  in  the  usual  manner.  If  the  greater 
altitude  is  within  the  limits  of  circummeridian  altitudes,  it  will 
of  course  be  reduced  by  (284). 

The  chief  objection  to  this  method  is  that  the  computation 
must  be  repeated  when  the  assumed  latitude  is  much  in  error. 
It  can  also  be  shown  that  unless  the  observations  are  taken  as 
near  to  the  meridian  as  we  have  above  supposed,  the  computed 
value  of  the  latitude  may  in  certain  peculiar  cases  be  more  in 
error  than  the  assumed  value,  so  that  successively  computed 
values  will  more  and  more  diverge  from  the  truth.  The  methods 
referred  to  in  the  preceding  articles  are,  therefore,  generally  to 
be  preferred. 

212.  The  latitude  may  also  be  found  from  two  altitudes  by 
the  simple  method  proposed  by  Captain  SUMNER,  for  which  see 
Chapter  VIIL 


BY    CHRONOMETERS.  317 


CHAPTER  VII. 

FINDING   THE    LONGITUDE    BY  ASTRONOMICAL   OBSERVATIONS. 

213.  THE  longitude  of  a  point  on  the  earth's  surface  i-s  the 
angle  at  the  pole  included  between  the  meridian  of  the  point 

.and  some  assumed  first  meridian.  The  difference  of  longitude 
of  any  two  points  is  the  angle  included  by  their  meridians. 
These  definitions  have  been  tacitly  assumed  in  Art.  45,  where 
we  have  established  the  general  equation 

L  =  TQ  —  T  (382) 

in  which  (Art.  47)  T0  and  T  are  the  local  times  (both  solar  or 
both  sidereal)  reckoned  respectively  at  the  first  meridian,  and  at 
that  of  any  point  of  the  earth's  surface,  and  L  is  the  west 
longitude  of  the  point. 

As  an  astronomical  question,  the  'determination  either  of  an 
absolute  longitude  from  the  first  meridian,  or  of  a  difference  of 
longitude  in  general,  resolves  itself  into  the  determination  of 
the  difference  of  the  time  reckoned  at  the  two  meridians  at  the 
same  absolute  instant.*  The  various  methods  of  finding  the 
longitude  which  are  treated  of  in  this  chapter  differ  only  in  the 
mode  by  which  the  comparison  of  the  times  at  the  two  meridians 
is  effected. 

FIRST   METHOD. — BY   PORTABLE   CHRONOMETERS. 

214.  The  difference  of  longitude  between  two  places  A  and 
B  being  required,  let  a  chronometer  be  accurately  regulated  at 
A,  that  is,  let  its  correction  on  the  time  at  that  place  and  its 
daily  rate  be  determined  by  the  methods  of  Chapter  Y. ;  then 
let  the  chronometer  be  transported  to  B,  and  let  its  correction 


*  The  astronomical  difference  of  longitude  may  differ  from  the  geodetic  difference 
for  the  same  reason  that  the  astronomical  latitude  differs  from  the  geodetic,  Arts.  86 
and  100. 


318  LONGITUDE. 

on  the  time  at  that  place  be  determined  at  any  instant.  The 
time  reckoned  at  A  at  this  last  instant  is  also  known  from  the 
correction  and  rate  first  found,  provided  the  rate  has  not  changed 
in  transportation;  and  hence  the  difference  of  times  at  the  same 
absolute  instant,  and  consequently  the  difference  of  longitude, 
are  found. 
Let 

AT7,  <5T—  the  correction  and  rate  determined  at  A  at  the 

time  T,  by  the  chronometer, 

AT"  =  the  correction  determined  at  B  at  the  time 
T'  =  T  -\-  t,  t  being  the  interval  by  the  chro- 
nometer ; 

then',  at  the  instant  T  -|-  t  the  true  time 

at  .4  is          T  -\-t  +  *T+t.dT 
"  B  T  -f  t  -f  A  T' 

and  hence  the  difference  of  longitude  is 

L  =  &T  +t.5T—  AT'  (383) 

Thus,  the  longitude  is  expressed  as  the  difference  of  the  two 
chronometer  corrections  at  the  two  places  ;  and  the  absolute 
indications  of  the  chronometer  do  not  enter,  except  so  far  as 
they  may  be  required  in  determining  the  interval  with  which 
the  accumulated  rate  is  computed.  In  this  expression  d  J'is  the 
rate  in  a  unit  of  the  chronometer  (an  hour,  or  a  day,  solar  or  sidereal), 
and  T'  —  T  must  be  expressed  in  that  unit. 

EXAMPLE.  —  At  Greenwich,  May  5,  mean  noon,  a  mean  time 
chronometer  marks  237t  49'"  42s.  75,  and  its  rate  in  24  chronometer 
hours  has  been  found  to  be  gaining  2*.671.  At  Cambridge,  Mass., 
May  17,  mean  noon,  the  same  chronometer  marks  47t  34m  47*.28; 
what  is  the  longitude  of  Cambridge  ? 

We  have 


!T=May  4,  23*  49*  42*.75     ±T=  +  0»10"17'.25     dT  =  —  2».671 
T+t=     "    17,    4  34   47.28 

t=          12*    4*45™   4'.53  =  12d.198 

Hence 

t.ST=  -f-  0*    9'"44'.67 
A  T  =  —  4   34    47.28 
L  =  -|-  4   44    31  .95 


BY    CHRONOMETERS.  319 

NOTE.  —  It  is  proper  to  distinguish  whether  the  given  rate  is  the  rate  in  a  chrono- 
meter unit  or  in  a  true  unit  of  time;  although  the  difference  will  not  be  appreciable 
unless  the  rate  is  unusually  great.  If  the  rate  is  20'  in  24*  by  the  chronometer,  it  will 
be  20*  it  Cf.005  in  24*  of  solar  time. 

215.  When  the  chronometer  is  carried  from  point  to  point 
without  stopping  to  rate  it  at  each,  it  is  convenient  to  prepare  a 
table  of  its  correction  for  noon  of  each  day  at  the  first  station, 
from  which  the  correction  for  the  time  of  any  observation  at  a 
transient  station  may  be  found  by  simple  interpolation. 

After  reaching  the  last  station,  it  is  proper  to  re-determine  the 
rate,  which  will  seldom  agree  precisely  with  that  found  at  the 
first.  In  the  absence  of  any  other  data  affecting  the  rate,  we 
may  assume  that  it  has  changed  uniformly  during  the  whole 
time.  It  is  convenient  to  compute  the  longitudes  first  upon  the 
supposition  of  a  constant  rate,  and  then  to  correct  them  for  the 
variation  of  rate,  as  follows.  Let 


j  dT  =  the  correction  and  rate  at  the  time  T,  found  at 

the  first  station, 
d'T=t\\Q  rate   found  at  the  last  station  at  the  time 

T+n, 

and  put 


(384) 


then  x  is  the  increase  of  rate  in  a  unit  of  time.  If  an  observa- 
tion at  an  intermediate  station  is  taken  at  the  time  T  -\-  t,  we 
must  compute  the  accumulated  rate  for  the  interval  /,  which  is 
effected  by  multiplying  the  mean  rate  during  this  interval  by  the 
interval.  But,  upon  the  supposition  of  a  uniform  increase,  the 
mean  rate  from  the  time  jTto  the  time  T  -{-  t  is  the  rate  at  the 
middle  instant  T -\-  \t,  and  this  rate  is  ST -j-  \tx.  Hence  the 
chronometer  correction  on  the  time  at  the  first  station  at  the 
instant  T  -f-  t  of  the  supposed  observation  is 

*r  +  t  (BT+  %tx)  =  ±T+t.dT+Wx  (385) 

A  longitude  assigned  to  an  intermediate  station  at  the  time 
T  -\-  t,  by  employing  the  original  rate  dT,  will  therefore  require 
the  correction  -f-  \t*x,  observing  always  the  algebraic  signs  of  x 
and  the  longitude. 


320  LONGITUDE. 

If  a  number  m  of  chronometers  have  been  employed,  and  each 
determination  of  a  longitude  is  the  mean  of  the  m  values  which 
they  have  severally  given,  the  longitude  assigned  upon  the  sup- 
position of  constant  rates  is  to  be  corrected  by  the  quantity 

V       x,  -f  x,  +  xa  +  &c.  +  xm 
2  X 

in  which  #!,  x2,  &c.  are  the  increments  of  the  rates  of  the  several 
chronometers  in  a  unit  of  time.     If  then  we  put 

s  =  the  sum  of  all  the  total  increments  during  the  whole 
interval  n,  or  the  sum  of  the  values  of  d'T —  dTfor 
the  several  chronometers, 


2mn 
we  shall  have 

Correction  of  a  longitude  at  a  time  T  -f  t  =  t-.q          (386) 

EXAMPLE.* — In  a  voyage  between  La  Guayra  and  Carthagena, 
calling  on  the  way  at  Porto  Cabello  and  Curagoa,  the  following 
observations  having  been  made,  the  relative  longitudes  are  re- 
quired. 

By  observations  at  La  Guayra  on  May  22  and  28,  the  cor- 
rections and  rates  of  chronometers  F,  M,  and  P  at  the  mean 
epoch  May  24^.885  were  as  follows : 

AT7  6T 

Chron.  F.          -  4»  33-    7'.80  -f  0'.77 

M.          -40    17.40  -4.54 

P.          -  5     9    43 .70  -  1 .47 

On  arrival  at  Porto  Cabello,  the  corrections  on  the  mean  time 
at  that  place  on  June  5t?.870  were  ascertained  to  be — 

F,  __  4*  37-  15..80 
M.  —45  31.28 
P.  -  5  14  13  .38 

At  Curagoa  the  corrections  on  June  12^.890  were — 

*  SHADWELL,  Notes  on  the  Management  of  Chronometers,  p.  111. 


BY    CHRONOMETERS.  321 


F.  —  4*  40-  59'.20 
M.  -49  55.53 
P.  -5  18  3.24 

And  finally,  at  Carthagena,  observations  on  the  25th  and  29th 
of  June  gave  the  corrections  and  rates  at  the  mean  epoch  June 
27d.O  as  follows: 

A'7*  6'T 

F.          -  5*    7-  238.55  -f  0-.85 

M.        —4   37    47.98  -5.90 

P.          -  5   44    34  .42  +  0  .30 

Employing  the  rates  found  at  La  Guayra,  the  corrections  of  the 
chronometers  on  June  5^.870  at  Porto  Cabello  (for  which  we 
have  t  =  11*.  985),  and  the  resulting  difference  of  longitude, 
are,  by  formula  (383),  are  as  follows  : 

±T  -j-  t.  6T  P.  Cabello  —  La  Guayra. 

F.  -  4*  32-  58'.57  +  4-  17*.23 

M.          -41    11.81  19.47 

P.          -5   10      1.32  12.06 

Mean  +  4    16.25 

With  the  same  rates,  we  have  on  June  12.890  at  Cura$oa  (for 
t  =  19rf.005)  the  corrections  and  the  corresponding  difference  of 
longitude,  as  follows: 

A  T  4-  *  .  6  T  Curasoa—  La  Guayra. 

F.          -4A32-53M7  +8-    6-.03 

M.          -41   43.68  8    11.85 

P.          -4   10    11.64  7    51  .60 

Mean  -f-  8      3  .16 

With  the  same  rates,  we  have  on  June  27d  at  Carthagena  (for 
t  =  33d.  115)  the  corrections  and  the  corresponding  difference  of 
longitude,  as  follows: 


F. 

M. 
P. 

VOL.  I.— 21 


A?7-!-  t.6T 
—  4*  32m  42-.30 

Carthagena  —  La  Gttayra. 
-f-  34-  41-.25 

—  4 

2 

47 

.74 

35 

0 

.24 

—  5 

10 

32 

.38 

34 

2 

.04 

Mean  -f  34 

34 

.51 

322  LONGITUDE. 

Now,  to  correct  these  results  for  the  changes  in  the  rates  of 
the  chronometers,  we  have,  in  the  interval  n  =  33.115, 

6'T—  6T 

F.  +  (K08 

M.  -  1  .36 

P.  +  1.77 

s  =  -f  0  .49 

and,  consequently, 

q  =  -  ±W®  -  -*  +  0-.002466' 
2  X  3  X  33.115 

Applying  the  correction  t2q  to  the  several  results,  the  true 
differences  of  longitude  from  La  Guayra  are  found  as  follows: 

Approx.  diff.  long.                         P.q  Corrected  diff.  long. 

P.  Cabello         +    4-  16-.25  -f  0'.35  -f    4-  16-.60 

Curacoa            -f    8      3  .16  -f  0  .89  +84  .05 

Carthagena      +  34    34  .51  +  2  .70  -f  34    37  .21 

But  it  is  usually  preferable  to  carry  out  the  result  by  each 
chronometer  separately,  in  order  to  judge  of  the  weight  to  be 
attached  to  the  final  mean  by  the  agreement  of  the  several  indi- 
vidual values.  For  this  purpose  we  have  here,  by  the  formula 
(384),  for  n  =  33.115, 

»« 

F.          +  0.00121 
M.  -  0.02054 

P.          -f  0.02673 

and  hence  the  correction  J  t2  .  x  is,  for  the  several  cases,  as  follows  : 

P.  Cabello.  Ctm^oa.1  •>                 Carthagena. 

F.          +  OM7  -f  0-.44  +    1-.32 

M.           -2.95  -7.41  -22.52 

P.           +  3  .84  +9  .65  -f  ^9  .31 

Applying  these  corrections  severally  to  the  above  approximate 
results,  we  have,  for  the  differences  of  longitude  from  La  Guayra, 


P.  Cabello. 

C 

ura9oa. 

Carthagena. 

F. 

+  4-  17-.40 

+ 

8*  6«.47 

-|-  34W  42'.57 

M. 

16.52 

4.44 

37.72 

P 

15.90 

1.25 

31.35 

Means  +  416  .6  4-8    405  -f  34    3721 

agreeing  precisely  with  the  corrected  means  found  above. 


iTY 


BY    CHRONOMETERS.  323 

If  the  chronometers  have  been  exposed  to  considerable 
changes  of  temperature,  the  proper  correction  may  be  intro- 
duced by  the  method  of  Art.  223. 

216.  Chronometric  expeditions  between  two  points. — Where  a  dif- 
ference of  longitude  is  to  be  determined  with  the  greatest 
possible  precision,  a  large  number  of  chronometers  are  trans- 
ported back  and  forth  between  the  extreme  points.  There  are 
two  classes  of  errors  of  chronometers  which  are  to  be  eliminated: 
1st,  the  accidental  errors,  or  variations  of  rate  which  follow  no 
law,  and  may  be  either  positive  or  negative ;  2d,  the  constant 
errors,  or  variations  of  rate  which,  for  any  given  chronometer, 
appear  with  the  same  sign  and  of  the  same  amount  when  the 
chronometer  is  transported  from  place  to  place ;  in  other  words, 
a  constant  acceleration,  or  a  constant  retardation,  as  compared 
with  the  rates  found  when  the  chronometer  is  at  rest.  The 
accidental  errors  are  eliminated  in  a  great  degree  by  employing 
a  large  number  of  chronometers,  the  probability  being  that  such 
errors  will  have  different  signs  for  different  chronometers.  The 
constant  errors  cannot  be  determined  by  comparing  the  rates  at 
the  two  extreme  points,  since  these  rates  are  found  only  when 
the  chronometer  is  at  rest ;  but  if  the  chronometers  are  trans- 
ported in  both  directions,  from  east  to  west  and  from  west  to 
east,  a  constant  error  in  their  travelling  rates  will  affect  the  differ- 
ence of  longitude  with  opposite  signs  in  the  two  journeys,  and 
will  disappear  when  the  mean  is  taken.  These  considerations 
have  given  rise  to  extensive  expeditions,  of  which  probably  the 
most  thoroughly  executed  was  that  carried  out  by  STRUVE,  in 
1843,  between  Pulkova  and  Altona.*  In  this  expedition  sixty- 
eight  chronometers  were  transported  eight  times  from  Pulkova 
to  Altona  and  back,  making  sixteen  voyages  in  all,  giving  the 
difference  of  longitude  between  the  centre  of  the  Pulkova  Obser- 
vatory and  the  Altona  Observatory  \h  2lm  32*. 527,  with  a  probable 
error  of  only  0'.039. 

Chronometric  expeditions  between  Liverpool  (England)  and 


*  Expedition  chronomttriquc  execute  par  ordre  de  Sa  Majestt  L'Empereur  Nicolas  /. 
pour  la  determination  de  la  longitude  geographique  relative  de  Vobservaloire  central  de 
Russie.  St.  Petersburg,  1844. 

For  an  account  of  the  carefully  executed  expedition  under  Professor  AIRY  to  deter- 
mine the  longitude  of  Valentia  in  Ireland,  see  the  Appendix  to  the  Greenwich 
Observations  of  1845. 


324  LONGITUDE. 

Cambridge  (U.  S.)  were  instituted  in  the  years  1849,  '50,  '51,  and 
'55  by  the  U.  S.  Coast  Survey,  under  the  superintendence  of 
Professor  A.  D.  BACHE.  The  results  of  the  expeditions  of  1349, 
'50,  and  '51,  discussed  by  Mr.  G.  P.  BOND,*  proved  the  necessity 
of  introducing  a  correction  for  the  temperature  to  which  the 
chronometers  were  exposed  during  the  voyages,  and  particular 
attention  was  therefore  paid  to  this  point  in  the  expedition  of 
1855,  the  details  of  which  were  arranged  by  Mr.  W.  C.  BOND. 
The  results  of  six  voyages, — three  in  each  direction, — according 
to  the  discussion  of  Mr.  G.  P.  BoND,f  were  as  follows : 

Longitude. 

Voyages  from  Liverpool  to  Cambridge    4*  32TO  31'.92 
"  "      Cambridge  to  Liverpool    4   32    31.75 

Mean    4   32    31.84 

with  a  probable  error  of  0'.19.  In  this  expedition  fifty  chrono- 
meters were  used.  The  greater  probable  error  of  the  result,  as 
compared  with  STRUVE'S,  is  sufficiently  explained  by  the  greater 
length  of  the  voyages  and  their  smaller  number. 

217.  The  following  is  essentially  STRUVE'S  method  of  conduct- 
ing the  expeditions  and  discussing  the  results. 

Before  embarking  the  chronometers  at  the  first  station  (A), 
they  are  carefully  compared  with  a  standard  clock  the  correction 
of  which  on  the  time  at  that  station-  has  been  obtained  with 
the  greatest  precision  by  transits  of  well-determined  stars.  (See 
Vol.  II.,  "  Transit  Instrument.")  Upon  their  arrival  at  the  second 
station  (JB),  they  are  compared  with  the  standard  clock  at  that 
station.];  From  these  two  comparisons  the  chronometer  correc- 
tions at  the  two  stations  become  known,  and,  if  the  rates  are 
known,  a  value  of  the  longitude  is  found  by  each  chronometer 
by  (383).  But  here  it  is  to  be  observed  that  the  rate  of  a  chro- 
nometer is  rarely  the  same  when  in  motion  as  when  at  rest.  It 
is  necessary,  therefore,  to  find  its  travelling  rate  (or  sea  rate,  as  it 
is  called  when  the  chronometer  is  transported  by  sea).  This 
might  be  effected  by  finding— -first,  the  correction  of  the  chrono- 

*  Report  of  the  Superintendent  of  the  U.  S.  Coast  Survey  for  1854,  Appendix  No.  42. 
f  Report,  of  the  Superintendent  of  the  U.  S.  Coast  Survey  for  1856,  p.  182. 
J  For  the  method  of  comparing  chronometers  and  clocks  with  the  greatest  pre- 
cision, see  Vol.  II. 


BY   CHRONOMETERS.  325 

meter  at  the  station  A  immediately  before  starting ;  secondly,  its 
correction  at  B  immediately  upon  its  arrival  there ;  and  thirdly, 
having,  without  any  delay  at  13,  returned  directly  to  A,  finding 
again  its  correction  there  immediately  upon  arriving.  The  dif- 
ference between  the  two  corrections  at  A  is  the  whole  travelling 
rate  during  the  elapsed  time,  and  this  rate  would  be  used  in 
making  the  comparison  with  the  correction  obtained  at  B,  and 
in  deducing  the  longitude  by  (383). 

But,  since  the  chronometer  cannot  generally  be  immediately 
returned  from  B,  its  correction  for  that  station  should  be  found 
both  upon  its  arrival  there  and  again  just  before  leaving,  and 
the  travelling  rate  inferred  only  from  the  time  the  instrument  is 
in  motion.  For  this  purpose,  let  us  suppose  that  we  have  found 

at  the  times  #,  t' ,  t",  t"', 

the  chron.  corrections     a,  b,  bf,  a', 

the  correction  a  at  the  station  A  before  leaving ;  b  upon  arriving 
at  -B;  b'  before  leaving  B ;  and  a'  upon  the  return  to  A.  The 
times  /,  t',  I",  t'" ,  being  all  reckoned  at  the  same  meridian,  if  we 
now  put 

m  =  the  mean  travelling  rate  of  the  chronometer  in  a  unit 

of  time, 
A  =  the  longitude  of  B  west  of  A, 

we  shall  have,  upon  the  supposition  that  the  mean  travelling 
rate  is  the  same  for  both  the  east  and  west  voyages, 

A  =  a  -f  m(t'  —  O  —  b 
X==af—m(tf"—tf'-)  —b' 

From  these  two  equations  the  two  unknown  quantities  m  and  A 
become  known.  Putting 

r=f—t  T"=t'"  —  t" 

we  find,  first, 

=  («*-a)-(y-» 

r  +  r" 

I 

in  which  the  numerator  evidently  expresses  the  whole  travelling 
rate,  and  the  denominator  the  whole  travelling  time.  Then, 
putting 


326  LONGITUDE. 

(<z)  =  a  +  ror  ^| 

we  have  H388) 

A  ==  (a)  -  b  \ 

in  which  (a)  is  the  interpolated  value  of  the  chronometer  correc- 
tion on  the  time  at  A,  for  the  same  absolute  instant  t'  to  which 
the  correction  b  on  the  time  at  J5  corresponds. 

EXAMPLE.  —  In  the  first  two  voyages  of  STRUVE'S  expedition 
between  Pulkova  and  Altona  in  1843,  the  corrections  of  the 
chronometer  "  Hauth  31"  were  found,  by  comparison  with  the 
standard  clocks  at  the  two  stations,  as  below.  The  dates  are  all 
in  Pulkova  time,  as  shown  by  one  of  the  chronometers  em- 
ployed in  the  comparison  : 

At  Pulkova  (A),  t   =.  May  19,  21*.54  a  =  -f-  0»    6-  38-.10 

"  Altona     (£),  t'  =     «     24,  22  .66  b  =  —  1    14    39  .92 

«  Altona     (£),  t"  =      «     26,  10  .72  b'  =  —  1    14    36  .77 

"  Pulkova  (A),  t"'=     «     31,    0.00  <z'=  +  0     7      9.58 

Hence 

T  =  5d    1M2  =  5d.047,         a'  —  a  =  +  31-.48 
r"  =  4   13  .28  =  4  ^553,          &'  —  &=+    3  .15 

=  2g*8 


5.047  -f-  4.553          9.6 

a  =  -f  0*  6m  38-.10 
mr=  -j-  14.89 

(a)  ^=4^0  6  52.99 

b  =  —  l   14   39.92 


l  =  (a)—b  =  +  l   21   32 .91 

* 

218.  In  the  above,  the  rate  of  the  chronometer  is  assumed  to 
be  constant,  and  the  problem  is  treated  as  one  of  simple  inter- 
polation. But  most  chronometers  exhibit  more  or  less  accelera- 
tion or  retardation  in  successive  voyages,  and  a  strict  interpola- 
tion requires  that  we  should  have  regard  to  second  differences. 
If  we  always  start  from  the  station  A,  as  in  the  above  example, 
using  only  simple  interpolation,  we  commit  a  small  error,  which 
always  affects  the  longitude  in  the  same  way  so  long  as  the 
variation  of  the  chronometer's  rate  preserves  the  same  sign. 
But  if  we  commence  the  next  computation  with  the  station  B, 


BY   CHRONOMETERS.  327 

so  that  the  two  chronometer  corrections  at  A  are  intermediate 
between  the  two  at  J3,  then  the  error  in  the  longitude  will  have 
a  different  sign,  and  the  mean  of  the  two  values  of  the  longitude 
will  be,  partially  at  least,  freed  from  the  influence  of  the  acce-X 
leration  or  retardation.  To  show  this  more  clearly  under  an 
algebraic  form,  let  us  suppose  that  we  have,  omitting  the  inter- 
vals of  rest  at  the  two  stations, 

at  the  times  t,  t,  t",  t", 

the  chron.  corrections      a,  b,  a',  b', 

intervals  T,  r',  T", 

and  that 

IJL  —  daily  rate  of  the  chronometer  at  the  time  #, 
2/9  =  the  daily  acceleration  of  the  rate  /JL  after  the  time  £, 

the  true  values  of  the  four  corrections,  observing  that  b  and  b' 
refer  to  the  meridian  of  J3,  will  be,  according  to  the  law  of  uni- 
formly accelerating  motion, 

a  =  a 

b  =  a  +  p.r  +  /9r2  —  A 

a'^a  +  M(r  +  r')+/9(r-f-O2 

b>  =  a  +  f*(r  +  r'  +  r")  +./9(r  +  *  +  T")'  -  * 

If  now  we  find  the  value  of  (a)  corresponding  to  b  (that  is,  for 
the  time  t')  by  simple  interpolation  between  the  values  of  a 
and  a',  we  have 


from  which  we  obtain  the  erroneous  longitude 

X  =  (a)  —  b  =  A  +  /9rr' 

Hence  the  error  in  the  longitude,  by  simple  interpolation  and 
commencing  with  the  station  A,  is  dX'  =  /9rrr. 

In  the  next  place,  if  we  commence  at  the  station  jB,  with  the 
correction  6,  employing  simple  interpolation  between  6  and  6', 
to  find  the  correction  (6)  for  the  time  t"  corresponding  to  a',  we 
have 


328  LONGITUDE. 


-  a  +  M(r  +  r')  +  /?02+  2T/  +  r"  +  r'r")  -  I 

and  we  find  the  erroneous  longitude 

A"  =  a'  —  (6)  =  A 


Hence  the  error  by  simple  interpolation,  commencing  with  the 
station  B,  is  dk"  =  —  /9rV  ;  and  the  error  in  the  mean  of  the 
two  longitudes  is 


an  error  which  disappears  altogether  when  the  intervals  r  and  r" 
are  equal.  Since  the  voyages  are  of  very  nearly  equal  duration, 
it  follows  that  by  computing  the  longitude,  as  proposed  by 
STRUVE,  commencing  alternately  at  the  two  stations,  the  final 
result  will  be  free  from  the  effect  of  any  regular  acceleration  or 
retardation  of  the  chronometers. 

EXAMPLE.  —  From  the  "Expedition  Chronometrique"  we  take 
the  following  values  for  the  chronometer  "  Hauth  31,"  being 
the  combination  next  following  after  that  given  in  the  example 
of  the  preceding  article,  commencing  now  with  the  station  J5,  or 
Altona  : 


At  Altona     (£),  t   =  May  26,  10».72  b  =  —  lh  14-  36'.77 

«   Pulkova  (A),  t  =     "     31,    0  .00  a  =  +  0     7      9  .58 

"   Pulkova  (4),  t"  =  June  3,    5.62  a'=  +  0     7    19.36 

"Altona     (B\t"=     "      7,20.52  b'=—  114      0.35 

Here 

r  =  4<*  13».28  =  4d.553  V  —  b  =  +  36«.42 

T"=4   14.90  =  4.621  a'  —  a=  -}-    9.78 

3G>.42-9.78       26-64 


4.553  -f-  4.621       9.174 

b  =  —  1*  14™  36«.77 
mr  =  _  +  13.22 
(b)  =  —  1  14  23.55 

a  =  -f  0  7  9  .58 
A  =  a  _  (^  =  -|-  1  21  33.13 

'The  mean  of  this  result  and  that  of  Art.  217  is  A  =  1*  21m  33'.02. 


BY   CHRONOMETERS.  329 

219.  Relative  weight  of  the  longitudes  determined  in  different  voyages 
by  the  same  chronometer. — From  the  above  it  appears  that  the 
problem  of  finding  the  longitude  by  chronometers  is  one  of 
interpolation.  If  the  irregularities  of  the  chronometer  are 
regarded  as  accidental,  the  mean  error  of  an  interpolated  value 
of  the  correction  may  be  expressed  by  the  formula* 


where  r  and  r'  have  the  same  signification  as  in  the  preceding 
article,  and  e  is  the  mean  (accidental)  error  in  a  unit  of  time. 
The  weight  of  such  an  interpolated  value  of  the  correction,  and, 
therefore,  also  the  weight  of  a  value  of  the  longitude  deduced 
from  it,  is  inversely  proportional  to  the  square  of  this  error,  and 
may,  therefore,  be  expressed  under  the  form 


where  k  is  a  constant  arbitrarily  taken  for  the  whole  expedition, 
so  as  to  give  p  convenient  values,  since  it  is  only  the  relative 
weights  of  the  different  voyages  which  are  in  question. 

But  if  the  chronometer  variations  are  no  longer  accidental, 
but  follow  some  law  though  unknown,  a  special  investigation 
may  serve  to  give  empirically  a  more  suitable  expression  of  the 
weight  than  the  above.  Thus,  according  to  STRUVE'S  investiga- 
tions in  the  case  of  certain  clocks,  the  weight  of  an  interpolated 
value  of  the  correction  for  these  clocks  could  be  well  expressed 
by  the  formulaf 


But  even  this  expression  he  found  could  not  be  generally  applied  ; 
and  he  finally  adopted  the  following  form  for  the  chronometric 
expedition  : 

(389) 


in  which  T  is  the  duration  of  an  entire  voyage,  including  the 

*  See  Vol.  II.,  "Chronometer." 
f  Expedition  Chron.,  p.  102. 


330  LONGITUDE. 

time  of  rest  at  one  of  the  stations,  r,  r"  are  the  travelling  times 
of  the  voyage  to  and  from  a  station,  and  K  is  an  arbitrary 
constant. 

Although  this  is  hut  an  empirical  formula,  it  represents  well 
the  several  conditions  of  the  problem.  For,  first,  the  weight  of 
a  resulting  longitude  must  decrease  as  the  length  of  the  voyage 
increases ;  and,  second,  it  must  become  greater  as  the  difference 
between  the  two  travelling  times  r,  r"  decreases,  since  (as  is 
shown  in  Yol.  II.,  "  Chronometer")  an  interpolated  value  of  a 
clock  correction  is  probably  most  in  error  for  the  middle  time 
between  the  two  instants  at  which  its  corrections  are  given. 

220.    Combination   of  results   obtained  by   the  same   chronometer, 

according  to  their  iveights. — Let  //,  /",  X" be  the  several  values 

of  the  longitude  found  by  the  same  chronometer,  according  to 

the  method  of  Arts.  217  and  218 ;    and  p',  p",  p'" their 

weights  by  formula  (389)  (or  any  other  formula  which  may  be 
found  to  represent  the  actual  condition  of  the  voyages) ;  then, 
according  to  the  method  of  least  squares,  the  most  probable 
value  of  the  longitude  by  this  chronometer  is 

= //  +  ?";."  +  ,/»*»  + 

ff  +  p"  +  p'"  + 

and  if  the  difference  between  this  value  and  each  particular 
value  be  found,  putting 

A'  —  L  =  v',  X'  —  L  =  v",  /"  —  L  =  v'",  (fee. 

n  =  the  number  of  values  of  A, 
s  =  the  mean  error  of  L, 
r  —  the  probable  error  of  L, 

then  we  shall  have 


=  J       [^7]r  r  =  0.6745  e  (391) 

X   n  —  1 


where  \_p~\  denotes  the  sum  of  p',  pff,  &c.,  and  [pvv]  the  sum  of 
p'v'v',  p"v"v",  &c. 

221.   Combination  of  the  results  obtained  by  different  chronometers, 
according  to  their  weights.  —  The  weights  of  the  results  by  different 


BY    CHRONOMETERS.  331 

chronometers  are  inversely  proportional  to  the  squares  of  theii 
mean  errors.  The  weight  P  of  a  longitude  L  will,  therefore,  be 
expressed  generally  by 


es 


in  which  k  is  arbitrary.     For  simplicity,  we  may  assume  k  =  1, 
and  then  by  the  above  value  of  s  we  shall  have 


Ipvv] 


If,  then,  _L',  -L",  .L"'  .....  are  the  values  found  by  the  several 
chronometers  by  (390),  P>  ',  P",  P"'  .....  their  weights  by  (392), 
the  most  probable  final  value  of  the  longitude  is 


P'  4-    P"    4-    P 
Then,  putting 


P"L"+P'"L'"  +  ..... 

—    ~ 


L'"-LQ=V"f    &c. 


-Q=,  -0=,  - 

N  =  the  number  of  values  of  L, 
E  =  the  mean  error  of  L0, 
E  =  the  probable  error  of  LQ1 
we  have 

*  =  0.6745.0  (394) 


222.  I  propose  to  illustrate  the  preceding  formulae  by  applying 
them  to  two  chronometers  of  STRUVE'S  expedition,  namely, 
"Dent  1774"  and  "Hauth  31."  In  the  following  table  the 
longitudes  found  by  beginning  at  Pulkova  are  marked  P,  those 
found  by  beginning  at  Altona  are  marked  A9  and  the  numeral 
accent  denotes  the  number  of  the  voyage.  The  weights  p  in  the 
second  column  are  as  given  by  STRUVE,  who  computed  them  by 
the  formula  (389),  taking  K=  34560  (the  intervals  T,  r,  r"  being 
in  hours),  which  is  a  convenient  value,  as  it  makes  the  weight  of 
a  voyage  of  nearly  mean  duration  equal  to  unity  ;  namely,  for 
T=  288*,  r  =  rf  =  120*.  If  we  express  T,  r,  T",  in  days,  we  take 


(24)' 


332  LONGITUDE. 

and  we  shall  have  STRUVE'S  values  of  p  by  the  formula 

60 


T\/rr" 


(395) 


Thus,  for  the  first  voyage,  we  have,  from  the  data  in  the  example 
of  Art.  217, 

T  =  tr"  —  t  =  lld  2*.46  =  1K103 

r  =  5d.047  r"  =   4d.553 

whence,  by  (395), 


P== 


60 


11.103  i/(5.047  X  4.553) 


=  1.13 


The  values  of  L1  and  L"  are  found  by  (390).  In  applying 
this  formula,  it  is  not  necessary  to  multiply  the  entire  longitudes 
by  their  weights,  but  only  those  figures  which  differ  in  the 
several  values.  Thus,  by  "Dent  1774"  we  have 

L'  =  1*  21-  30'  4-  2''51  X  1.10  +  2'.83  X  1-02  +  2».Q9  X  l.U  +  &c. 

1.10  +  1.02  -f-  1.14  +  &c. 
=  1*  21-  30'       2*.46 


Weight. 
J> 

Longitudes  by 
Chronometer 
Dent  1774. 

V 

pvv 

Longitudes  by 
Chronometer 
Hauth  31. 

V 

pvv 

pi 

.13 

1A  21"*  32'.91 

-f  (K30 

0.102 

A» 

.06 

33.13 

-f  0.52 

0.287 

pa 

.10 

1*  21«  32'.51 

+  0'.05 

0.003 

33  .36 

-f  0.75 

0.619 

A» 

.02 

32.83 

+  0.37 

0.140 

33.12 

-f  0.51 

0.265 

piii 

.14 

32.09 

—  0.37 

0.156 

32.55 

—  0.06 

0.004 

Aui 

.05 

32.25 

—  0  .21 

0.046 

31  .56 

—  1  .05 

1.158 

piv 

.19 

31.69 

—  0.77 

0.706 

32.70 

+  0.09 

0.010 

Aiv 

0.96 

32.77 

-f-  0.31 

0.092 

34.16 

-f  1.55 

2.306 

pv 

.09 

32.79 

+  0.33 

0.119 

32.23 

-  0.38 

0.157 

Av 

0.80 

32.54 

+  0.08 

0.005 

31.65 

—  0  .96 

0.737 

pvi 

.00 

32.94 

+  0.48 

0.230 

33.38 

-f  0.77 

0.593 

Avi 

.10 

31  .93 

—  0.53 

0.309 

31  .97 

-  0  .64 

0.451 

pvii 

1.20 

32.34 

—  0.12 

0.017 

83  .16 

+  0.55 

0.363 

Avii 

1.09 

32.95 

-f  0.49 

0.262 

%        81  .78 

—  0  .83 

0.751 

pvHI 

0.76 

31  .86 

—  0.60 

0.274 

30  .92 

—  1  .69 

2.171 

Avni 

0.41 

33  .77 

+  1  .31 

0.704 

L'  =  1»  21»  32'.46               [pvv]  ==  3.063 

//'  =  !*  21™  32'.  (11     [pvv]=  9.974 

n  =  14              |>]  =  13.91 

n  =  15                [/>]  =  15.69 

P'      13X  13.91 

14X15.69 

3.063 

9.974 

.6745 

r"        df6745=±:<M4 

BY    CHRONOMETERS.  333 

Combining  these  two  results,  we  have,  by  (393), 

£.=  1.  21"  32-  +  °''46  X  ™  +  ^61  X  22  =  1'  21"  32-.501 

with  the  probable  error,  by  (394), 

R  =  ±  (K067 

This  agrees  very  nearly  with  the  final  result  from  the  sixty-eight 
chronometers. 

223.  In  the  preceding  method,  the  sea  rate  is  inferred  from 
two  comparisons  of  the  chronometer  made  at  the  same  place 
before  and  after  the  voyages  to  and  from  the  second  place ;  and 
the  correction  of  the  chronometer  on  the  time  of  the  first  place 
at  the  instant  when  it  is  compared  with  the  time  of  the  second 
place  is  interpolated  upon  the  theory  that  the  rate  has  changed 
uniformly.  This  theory  is  insufficient  when  the  temperature  to 
which  the  chronometer  is  exposed  is  not  constant  during  the 
two  voyages,  or  nearly  so.  I  shall,  therefore,  add  the  method 
of  introducing  the  correction  for  temperature  in  cases  where 
circumstances  may  seem  to  require  it. 

According  to  the  experience  of  M.  LIEUSSON,  the  rate  m  of  a 
chronometer  at  a  given  temperature  &  may  be  expressed  by  the 
formula  (see  Vol.  II.,  "  Chronometer") 

m  =  m0  -f  k  (0  —  #0)2  —  k't  (396) 

in  which  $0  is  the  temperature  for  which  the  balance  is  compen- 
sated, m0  the  rate  determined  at  that  temperature  at  the  epoch 
t  =  0,  t  being  the  time  from  this  epoch  for  which  the  rate  m  is 
required,  k  the  constant  coefficient  of  temperature,  and  kf  that 
of  acceleration  of  the  chronometer  resulting  from  thickening  of 
the  oil  or  other  gradual  changes  which  are  supposed  to  be  pro- 
portional to  the  time. 

It  is  evident  that,  since  every  change  of  temperature  produces 
an  increase  of  w,  the  term  k(&  —  $0)2  will  not  disappear  even  when 
the  mean  value  of  $  is  the  same  as  $0.  It  is  necessary,  therefore, 
to  determine  the  sum  of  the  effects  of  all  the  changes.  Let  us, 
therefore,  determine  the  accumulated  rate  for  a  given  period  of 
time  T.  Let  m0  be  tho  rate  at  the  middle  of  this  period,  in  which 
case  we  have  in  tho  formula  t  =  0.  A  strict  theory  requires  that 


334  LONGITUDE. 

we  should  know  the  temperature  at  every  instant  ;  but,  in  default 
of  this,  let  us  assume  that  the  period  r  is  divided  into  sufficiently 
small  intervals,  and  that  the  temperature  is  observed  in  each. 
Let  us  suppose  n  equal  intervals  whose  sum  is  r,  and  denote  the 
observed  values  of  &  by  #<l>,  #<2>,  &*>  ____  #<">.  The  rate 


in  the  1st  interval  is  [m0  -f-  k  (#(1)  —  £0)2]  X  - 


&c.  &c. 

in  the  nth  interval  is  [mfl  -f  k  (#(H)  —  #0)2]  X  - 

and  the  accumulated  rate  in  the  time  r  is  the  sum  of  these 
quantities, 


where  2n(&  —  $0)2  denotes  the  sum  of  the  n  values  of  (d  —  $0)2. 
To  make  this  expression  exact,  we  should  have  an  infinite  number 

of  infinitesimal  intervals,  or  we  must  put-  -dr,  and  substitute 

the  integral  sign  J  for  the  summation  symbol  2:  thus,  the  exact 
expression  for  the  whole  rate  in  the  time  r  is 

m0r  + 

This  integral  cannot  be  found  in  general  terms,  since  &  cannot 
be  expressed  as  a  function  of  r ;  but  we  can  obtain  an  approxi- 
mate expression  for  it,  as  follows.  Let  ^  be  the  mean  of  all  the 
observed  values  of  & ;  then  we  have 


in  which  ^  —  #0  is  constant,  and,  therefore,  for  n  values  we  have 
2n  (#t —  ^2  =  n  (t?t —  $0)2.  Moreover,  since  ^  is  the  mean  of  all 
the  values  of??,  we  have  £n(&  —  $0  =  0,  and,  consequently,  also 
Jn2  (^  —  #0)  (*  —  '^)  =  2  (^—  ^0)  ^n  (^  —  ^)  =  0 ;  and  the  above 
expression  becomes 


BY   CHRONOMETERS.  335 

Hence,  also, 

Sn  (*  -  'V  £  =  *  (*1  ^  *<02  +   3,  (*  ~  *l)'  £ 

or,  for  an  infinite  value  of  ft, 

r  (* — #0)s  ^ = T  (#1 — 'U2 + J7  (* — #i)2  <fr 

•/  0  «/  0 

Thus,  the  required  integral  depends  upon  the  integral  f  r  (# — ^)2  c?r, 

which  may  be  approximately  found  from  the  observed  values  of 
$  by  the  theory  of  least  squares.  For,  if  we  treat  the  values  of 
$  —  $!  as  the  errors  of  the  observed  values  of  $,  and  denote  the 
mean  error  (according  to  the  received  acceptation  of  that  term 
in  the  method  of  least  squares)  by  e,  we  have 

^  (398) 


in  which  n  is  the  actual  number  of  observed  values  of  $.  If  we 
assume  that  a  more  extended  series  of  values,  or  indeed  an  infi- 
nite series,  would  exhibit  the  same  mean  error  (which  will  be 
the  more  nearly  true  the  greater  the  number  n\  we  assume  the 
general  relation 

^y  (*  —  #i)2  =(N—V)  e* 

^t 

in  which  N  is  any  number.     Hence,  also,  //• 

^(9_,yz=Te2£izi    I  Library 

1    N 
and,  making  N  infinite, 


'»-Wdr=Te*  (399) 

Substituting  this  value,  the  formula  (397)  becomes 


or  [m0  -j-  k  (tfj  —  #0)2  -f-  ke2]r  (400) 


from  which  it  appears  that  ra0-f-  k(^l—  #0)2+  A-e2  is  the  mean  rate 
in  a  unit  of  time  for  the  interval  r,  m0  being  the  rate  at  the 
middle  of  the  interval  for  a  temperature  &  =  $0.-  For  any  subse- 
quent interval  r',  we  must,  according  to  (396),  replace  m0  by 
ra0 —  &r£,  £  being  the  interval  from  the  middle  of  r  to  the  middle 
of  r'. 


336  LONGITUDE. 

Now,  let  us  suppose  that  the  chronometer  correction  is  obtained 
hy  astronomical  observations*  at  the  station  A,  at  the  times  T^ 
and  jT2,  before  starting  upon  the  voyage,  and  again  after  reaching 
the  station  B,  at  the  times  T3  and  jT4,  these  times  being  all 
reckoned  at  the  same  meridian.  Let  av  «2,  #3,  #4,  be  the  observed 
corrections,  and  put 

T  __    rp  _  rp  rp   _     /  rp   __    rp  _     n 

j-a  —  J.  j  —  r,  ±  3  —  i  a  —   r  ,  j_  4  —  j.3  — 

so  that  T  and  r"  are  the  shore  intervals  and  r'  the  sea  interval. 
Let  the  adopted  epoch  of  the  rate  m0  be  the  middle  of  the  sea 
interval  r'  ;  then,  by  (400),  with  the  correction  k't,  the  accumu- 
lated rates  in  the  three  intervals  are 


+*(*.  -  *•)• 


+  a3  -  a,  =  [m0  +  It  (#/  -  *0)2  +  ^  ]  r'       (401) 

S'  \ 


in  which  $j,  ??/,  $/'  are  the  mean  temperatures  in  the  intervals 
r,  r',  r",  and  *,  er,  er/  are  found  by  the  formula  (398).  These 
three  equations  determine  the  three  unknown  quantities  m0,  A/, 
and  L  If  we  put 


^ 


f"  —       *~ "       3  I  fA  "  ,9  ^2  Z-P"2 

y  ^        -  K  (9t    -  - 1>0;  - 

we  have,  from  the  first  and  third  equations, 


which  substituted  in  the  second  equation  gives  L  If,  however, 
we  prefer  to  compute  the  approximate  longitude  without  con- 
sidering the  temperatures,  and  afterwards  to  correct  for  tempe- 
rature, we  shall  have 


TERRESTRIAL    SIGNALS.  337 

W__(a3_«s) +  (_^+_^_^  +  i*(T     r)r   \ 


These  formulae  apply  to  a  voyage  in  either  direction .;  but  in  the 
case  of  a  voyage  from  west  to  east  they  give  /  with  the  negative 
sign. 

The  term  J£'(r"  —  r)  rr  in  the  first  equation  of  (402)  will  not 
be  rigorously  obtained  if  the  temperatures  are  neglected;  but  it 
is  usually  an  insensible  term  in  practice,  as  r"  and  T  are  made 
as  nearly  equal  as  possible,  and  kf  is  always  very  small. 

In  combining  the  results  of  different  chronometers  employed 
in  the  same  voyage,  the  weight  of  each  may  be  assigned  accord 
ing  to  the  regularity  of  the  chronometer  as  determined  from  its 
observed  rates  from  day  to  day.* 

SECOND    METHOD. — BY    SIGNALS. 

224.  Terrestrial  Signals. — If  the  two  stations  are  so  near  to  each 
other  that  a  signal  made  at  either,  or  at  an  intermediate  station, 
can  be  observed  at  both,  the  time  may  be  noted  simultaneously 
by  the  clocks  of. the  two  stations,  and  the  difference  of  longitude 
at  once  inferred.  The  signals  may  be  the  sudden  disappearance 
or  reappearance  of  a  fixed  light,  or  flashes  of  gunpowder,  &c. 

If  the  places  are  remote,  they  may  be  connected  by  interme- 
diate signals.  For  example :  suppose  four  stations,  A,  J9,  (7,  -D, 
chosen  from  east  to  west,  the  first  and  last  being  the  principal 
stations  whose  difference  of  longitude  is  required.  At  the  in- 
termediate stations  J9,  C  let  observers  be  stationed  with  good 
chronometers  whose  rates  are  known.  Let  signals  be  made  at 
three  points  intermediate  between  A  and  J9,  B  and  (7,  C  and  7), 
respectively.  The  signals  must,  by  a  preconcerted  arrangement, 
be  made  successively,  and  so  that  the  observers  at  the  interme- 
diate stations  may  have  their  attention  properly  directed  upon 
the  appearance  of  the  signal.  If,  then,  at  the  first  signal  the 
observers  at  A  and  B  have  noted  the  times  a  and  b;  at  the 


*  Besides  the  papers  already  referred  to,  see  the  Report  of  the  Superintendent  of 
the  U.  S.  Coast  Survey  for  1857,  p.  314. 
VOL.  I.— 22 


338 


LONGITUDE. 


second  signal  the  observers  at  B  and  C  the  times  bf  and  c;  at 
the  third  signal  the  observers  at  C  and  D  the  times  c1-  and  d;  it 
is  evident  that  the  time  at  A  when  the  third  signal  is  made  is 
a  -f  (bf  —  b)  +  (cf—  c),  at  which  instant  the  time  at  -D  is  d:  hence 
the  difference  of  longitude  of  A  and  D  is 


=  a  -f  (V  —  b)  +  (Y  —  c)  — 


(403) 


and  so  on  for  any  number  of  intermediate  stations.  It  is  re- 
quired of  the  intermediate  chronometers  only  that  they  should 
give  correctly  the  differences  bf —  6,  cf —  <?,  for  which  purpose 
only  their  rates  must  be  accurately  known.  The  daily  rates  are 
obtained  by  a  comparison  of  the  instants  of  the  signals  on  suc- 
cessive days.  Small  errors  in  the  rates  will  be  eliminated  by 
making  the  signals  both  from  west  to  east  and  from  east  to 
west,  and  taking  the  mean  of  the  results. 

The  intervals  given  by  the  intermediate  chronometers  should, 
of  course,  be  reduced  to  sidereal  intervals,  if  the  clocks  at  the 
extreme  stations  are  regulated  to  sidereal  time. 

EXAMPLE. — From  the  Description  G-eometrique  de  la  France 
(PUISSANT).  On  the  25th  of  August,  1824,  signals  were  observed 
between  Paris  and  Strasburg  as  follows: 


Paris. 

A 
19*  6™  20«.3 


Intermediate  Stations. 

--^ 

B  C 


8   54    10.8 


9*  16771    0*.2 
9   30    37.8 


Strasburg. 
D 

19*  46-  51«.4 


The  correction  of  the  Paris  clock  on  Paris  sidereal  time  was 
—  36*.  2 ;  that  of  the  Strasburg  clock  on  Strasburg  sidereal  time  was 
-  27'.7.     The  chronometers  at  B  and  C  were  regulated  to  mean 
time,  and  their  daily  rates  were  so  small  as  not  to  be  sensible  in 
the  short  intervals  which  occurred. 
We  have 

b'—b=    4"  22'.6 

c'—c  =  U  37.6 
Mean  interval  =19  0  .2 
Eed.  to  sid.  int.  —  -f  3  .1 
Sid.  interval  =19  3  .3 


CELESTIAL    SIGNALS.  339 

Paris  clock  J9*    6M  2CS.3     Strasburg  clock         19*  46M  51«.4 

Correction  —  35  .2     Correction  27  .7 

Paris  sid.  time  19      5    44  .1     Strasburg  sid.  time  19   46    23  .7 

Sid.  interval  -f-  19      3  .3 

Paris  sid.  time  of  the  ) 

last  signal  }  19    24    47  A 

Strasburg  do.  19   46    23.7 

A  *a±    0*  2  lm  3G'.3 


In  the  survey  of  the  boundary  between  the  United  States  and 
Mexico,  Major  W.  II.  EMORY,  in  1852,  employed  flashes  of  gun- 
powder as  signals  in  determining  the  diff.  of  long,  of  Frontera 
and  San  Elciario.* 

The  signals  may  be  given  by  the  heliotrope  of  GAUSS,  by  which 
an  image  of  the  sun  is  reflected  constantly  in  a  given  direction 
towards  the  distant  observer.  Either  the  sudden  eclipse  of  the 
light,  or  its  reappearance,  may  be  taken  as  the  signal ;  the 
eclipse  is  usually  preferred. 

Among  the  methods  by  terrestrial  signals  may  be  included 
that  in  which  the  signal  is  given  by  means  of  an  electro-tele- 
graphic wire  connecting  the  two  stations;  but  this  important 
and  exceedingly  accurate  method  will  be  separately  considered 
below. 

225.  Celestial  Signals. — Certain  celestial  phenomena  which  are 
visible  at  the  same  absolute  instant  by  observers  in  various  parts 
of  the  globe,  maybe  used  instead  of  the  terrestrial  signals  of  the 
preceding  article :  among  these  we  may  note — 

a.  The  bursting  of  a  meteor,  and  the  appearance  or  disappear- 
ance  of  a   shooting   star. — The  difficulty  of  identifying  these 
objects  at  remote   stations  prevents  the  extended  use  of  this 
method. 

b.  The  instant  of  beginning  or  ending  of  an  eclipse  of  the 
moon. — This  instant,  however,  cannot  be  accurately  observed, 
on  account  of  the  imperfect  definition  of  the  earth's  shadow.    A 
rude  approximation  to  the  difference  of  longitude  is  all  that  can 
be  expected  by  this  method. 

c.  The  eclipses  of  Jupiter's  satellites  by  the  shadow  of  that 
planet. — The   Greenwich  times   of  the  disappearance   of  each 


*  Proceedings  of  8th  Meeting  of  Am.  Association,  p.  64. 


340  LONGITUDE. 

satellite,  and  of  its  reappearance,  are  accurately  given  in  the 
Ephemeris :  so  that  an  observer  who  has  noted  one  of  these 
phenomena  has  only  to  take  the  difference  between  this  observed 
local  time  of  its  occurrence  and  the  Greenwich  time  given  in  the 
Ephemeris,  to  have  his  absolute  longitude.  With  telescopes  of 
different  powers,  however,  the  instant  of  a  satellite's  disappear- 
ance must  evidently  vary,  since  the  eclipse  of  the  satellite  takes 
place  gradually,  and  the  more  powerful  the  telescope  the  longer 
will  it  continue  to  show  the  satellite.  If  the  disappearance  and 
reappearance  are  both  observed  with  the  same  telescope,  the 
mean  of  the  results  obtained  will  be  nearly  free  from  this  error. 
The  first  satellite  is  to  be  preferred,  as  its  eclipses  occur  more 
frequently  and  also  more  suddenly.  Observers  who  wish  to 
deduce  their  difference  of  longitude  by  these  eclipses  should  use 
telescopes  of  the  same  power,  and  observe  under  the  same 
atmospheric  conditions,  as  nearly  as  possible.  But  in  no  case 
can  extreme  precision  be  attained  by  this  method. 

d.  The  occultations  of  Jupiter's  satellites  by  the  body  of  the 
planet. — The  approximate  Greenwich  times  of  the  disappearance 
behind  the  disc,  and  the  reappearance  of  each  satellite,  are  given 
in  the  Ephemeris.     These  predicted  times  serve  only  to  enable 
the  observers  to  direct  their  attention  to  the  phenomenon  at  the 
proper  moment. 

e.  The  transits  of  the  satellites  over  Jupiter's  disc. — The  ap- 
proximate Greenwich  times  of  "ingress"  and  "egress,"  or  the 
first  and  last  instants  when  the  satellite  appears  projected  on 
the  planet's  disc,  are  given  in  the  Ephemeris. 

/.  The  transits  of  the  shadows  of  the  satellites  over  Jupiter's 
disc. — The  Greenwich  times  of  "ingress"  and  "egress"  of  the 
shadow  are  also  approximately  given  in  the  Ephemeris. 

Among  the  celestial  signals  we  may  include  also" eclipses  of 
the  sun,  or  occultations  of  stars  and  planets  by  the  moon,  or, 
in  general,  the  arrival  of  the  moon  at  any  given  position  in  the 
heavens;  but,  in  consequence  of  the  moon's  parallax,  these 
eclipses  and  occultations  do  not  occur  at  the  same  absolute  in- 
stant for  all  observers,  and,  in  general,  the  moon's  apparent 
position  in  the  heavens  is  affected  by  both  parallax  and  refrac- 
tion. The  methods  of  employing  these  phenomena  as  signals, 
therefore,  involve  special  computations,  and  will  be  hereafter 
treated  of.  See  the  general  theory  of  eclipses,  and  the  method 
of  lunar  distances. 


BY    THE    ELECTRIC    TELEGRAPH.  341 

THIRD    METHOD.  —  BY   THE    ELECTRIC    TELEGRAPH. 

226.  It  is  evident  that  the  clocks  at  two  stations,  A  and  B, 
may  be  compared  by  means  of  signals  communicated  through 
an  electro-telegraphic  wire  which  connects  the  stations.  Sup- 
pose at  a  time  T  by  the  clock  at  J.,  a  signal  is  made  which  is 
perceived  at  B  at  the  time  T'  by  the  clock  at  that  station.  Let 
AT1  and  A  Tf  be  the  clock  corrections  on  the  times  at  these  sta- 
tions respectively  (both  being  solar  or  both  sidereal).  Let  x  be 
the  time  required  by  the  electric  current  to  pass  over  the  wire  ; 
then,  A  being  the  more  easterly  station,  we  have  the  difference 
of  longitude  X  by  the  formula 


Since  x  is  unknown,  we  must  endeavor  to  eliminate  it.  For 
this  purpose,  let  a  signal  be  made  at  B  at  the  clock  time  T"  , 
which  is  perceived  at  A  at  the  clock  time  T'"  ;  then  we  have 

A  p=  (  T'"  +  A  T'")  —  (  T"  +  A  T")  —  x  =  X2  —  x 

In  these  formulae  ^  and  ),2  denote  the  approximate  values  of  the 
difference  of  longitude,  found  by  signals  east-west  and  west-east 
respectively,  when  the  transmission  time  x  is  disregarded;  and 
the  true  value  is 


Such  is  the  simple  and  obvious  application  of  the  telegraph  to 
the  determination  of  longitudes;  but  the  degree  of  accuracy 
of  the  result  depends  greatly  —  more  than  at  first  appears  — 
upon  the  manner  in  which  the  signals  are  communicated  and 
received. 

Suppose  the  observer  at  A  taps  upon  a  signal  key*  at  an  exact 
second  by  his  clock,  thereby  producing  an  audible  click  of  the 
armature  of  the  electro-magnet  at  B.  The  observer  at  B  may 
not  only  determine  the  nearest  second  by  his  clock  when  he 
hears  this  click,  but  may  also  estimate  the  fraction  of  a  second; 
and  it  would  seem  that  we  ought  in  this  way  to  be  able  to  deter- 
mine a  longitude  within  one-tenth  of  a  second.  But,  before  even 
this  degree  of  accuracy  can  be  secured,  we  have  yet  to  eliminate, 
or  reduce  to  a  minimum,  the  following  sources  of  error: 

*  See  Vol.  II.,  "Chronograph,"  for  the  details  of  the  apparatus  here  alluded  to. 


342  LONGITUDE. 

1st.  The  personal  error  of  the  observer  who  gives  the  signal; 
2d.  The  personal  error  of  the  observer  who  receives  the  signal 

and  estimates  the  fraction  of  a  second  by  the  ear; 
3d.  The  small  fraction  of  time  required  to  complete  the  galvanic 

circuit  after  the  finger  touches  the  signal  key; 
4th.  The  armature  time,  or  the  time  required  by  the  armature  at 

the  station  where  the  signal  is  received,  to  move  through 

the  space  in  which  it  plays,  and  to  give  the  audible  click; 
5th.  The  errors  of  the  supposed  clock  corrections,  which  involve 

errors  of  observation,  and  errors  in  the  right  ascensions  of 

the  stars  employed. 

For  the  means  of  contending  successfully  with  these  sources 
of  error  we  are  indebted  to  our  Coast  Survey,  which,  under  the 
superintendence  of  Prof.  Bache,  not  only  called  into  existence 
the  chronographic  instruments,  but  has  given  us  the  most  effi- 
cient method  of  using  them.  The  "method  of  star  signals,"  as 
it  is  called,  was  originally  suggested  by  the  distinguished  astro- 
nomer Mr.  S.  C.  Walker,  but  its  full  development  in  the  form 
now  employed  in  the  Coast  Survey  is  due  to  Dr.  B.  A.  Gould. 

227.  Method  of  Star  Signals. — The  difference  of  longitude  be- 
tween the  two  stations  is  merely  the  time  required  by  a  star  to 
pass  from  one  meridian  to  the  other,  and  this  interval  may  be 
measured  by  means  of  a  single  clock  placed  at  either  station,* 
but  in  the  main  galvanic  circuit  extending  from  one  station  to 
the  other.  Two  chronographs,  one  at  each  station,  are  also  in 
the  circuit,  and,  when  the  wires  are  suitably  connected,  the  clock 
seconds  are  recorded  upon  both.  A  good  transit  instrument  is 
carefully  mounted  at  each  station. 

When  the  star  enters  the  field  of  the  transit  instrument  at  A 
(the  eastern  station),  the  observer,  by  a  preconcerted  signal  with 
his  signal  key,  gives  notice  to  the  assistants  at  both  A  and  B, 
who  at  once  set  the  chronographs  in  motion,  and  the  clock  then 
records  its  seconds  upon  both.  The  instants  of  the  star's  tran- 
sits over  the  several  threads  of  the  reticule  are  also  recorded 
upon  both  chronographs  by  the  taps  of  the  observer  upon  his 
signal  key.  When  the  star  has  passed  all  the  threads,  the  ob- 


*  The  clock  may,  indeed,  be  at  any  place  which  is  in  telegraphic  connection  with 
the  two  stations  whose  difference  of  longitude  is  to  be  found. 


BY   THE    ELECTRIC   TELEGRAPH.  343 

server  indicates  it  by  another  preconcerted  signal,  the  chrono- 
graphs are  stopped,  and  the  record  is  suitably  marked  with  date* 
name  of  the  star,  and  place  of  observation,  to  be  subsequently 
identified  and  read  off  accurately  by  a  scale.  When  the  star 
arrives  at  the  meridian  of  B,  the  transit  is  recorded  in  the  same 
manner  upon  both  chronographs. 

Suitable  observations  having  been  made  by  each  observer  to 
determine  the  errors  of  his  transit  instrument  and  the  rate  of 
the  clock,  let  us  put 

Tl  =  the  mean  of  the  clock  times  of  the  eastern  transit  of 

the  star  overall  the  threads,  as  read  from  the  chrono- 

graph at  A, 

T3  =  the  same,  as  read  from  the  chronograph  at  B, 
TI  =  the  mean  of  the  clock  times  of  the  western  transit  of 

the  star  over  all  the  threads,  as  read  from  the  chrono- 

graph at  A, 

TJ  =  the  same  as  read  from  the  chronograph  at  _B, 
Cj  e'  =  the  personal  equations  of  the  observers  at  A  and  B 

respectively, 
T,T'  =  the  corrections  of  Tt  and  T/  (or  of  T3and  Ta')  for 

the  state  of  the  transit  instruments  at  A  and  B,  or 

the  respective  "reductions  to  the  meridian"  (Vol.  II., 

Transit  Inst.), 

dT  =  the  correction  for  clock  rate  in  the  interval  T/  —  T^ 
x  =  the  transmission  time  of  the  electric  current  between 

A  and  B, 
X  =  the  difference  of  longitude; 

then  it  is  easily  seen  that  we  have,  from  the  chronographic 
records  at  A, 


X  =  2\f  +  JT+  r'  +  e'  —  x  —  (T,  +  r  +  e) 
and  from  the  chronographic  records  at  B, 


and  the  mean  of  these  values  is 

*  -  H(Zy  +  ZV)  +  <]  -  [K?\  +  TJ  +  <1  +  T+  e'~e    (404> 
which  we  may  briefly  express  thus  : 

A  =  ^  -f  e'  —  e 


344  LONGITUDE. 

in  which 

^=  the  approximate  difference  of  longitude  found  by  the 
exchange  of  star  signals,  when  the  personal  equations 
of  the  observers  are  neglected. 

This  equation  would  be  final  if  ef  —  e,  or  the  relative  personal 
equation  of  the  observers,  were  known  :  however,  if  the  observers 
now  exchange  stations  and  repeat  the  above  process,  we  shall 
have,  provided  the  relative  personal  equation  is  constant, 

A  =  la  -f  e  —  ef 

in  which  Xz  is  the  approximate  difference  of  longitude  found  as 
before  ;  and  hence  the  final  value  is 


I  have  not  here  introduced  any  consideration  of  the  armature 
time,  because  it  affects  clock  signals  and  star  signals  in  the  same 
manner;  and  therefore  the  time  read  from  the  chronographic 
fillet  or  sheet  is  the  same  as  if  the  armature  acted  instanta- 
neously.* It  is  necessary,  however,  that  this  time  should  be 
constant  from  the  first  observation  at  the  first  station  to  the 
last  observation  at  the  second,  and  therefore  it  is  important  that 
no  changes  should  be  made  in  the  adjustments  of  the  apparatus 
during  the  interval. 

As  the  observer  has  only  to  tap  the  transits  of  the  star  over 
the  threads,  the  latter  may  be  placed  very  close  together.  The 
reticules  prepared  by  Mr.  W.  WURDEMANN  for  the  Coast  Survey 
have  generally  contained  twenty-five  threads,  in  groups  or  "tal- 
lies" of  five,  the  equatorial  intervals  between  the  threads,  of  a 
group  being  2s.  5,  and  those  between  the  groups  5*  ;  with  an  ad- 
ditional thread  on  each  side  at  the  distance  of  10*  for  use  in  ob- 
servations by  "eye  and  ear."  Except  when  clouds  intervene 
and  render  it  necessary  to  take  whatever  threads  may  be  avail- 
able, only  the  three  middle  tallies,  or  fifteen  threads,  are  used. 
The  use  of  more  has  been  found  to  add  less  to  the  accuracy  of  a 

*  Dr.  B.  A.  GOULD  thinks  that  the  armature  time  varies  with  the  strength  of  the 
battery  and  the  distance  (and  consequent  weakness)  of  the  signal;  being  thus  liable 
to  be  confounded  with  the  transmission  time.  The  effect  upon  the  difference  of 
longitude  will  be  inappreciable  if  the  batteries  are  maintained  at  nearly  the  same 
strength. 


BY    THE    ELECTRIC    TELEGRAPH.  345 

determination  than  is  lost  in  consequence  of  the  greater  fatigue 
from  concentrating  the  attention  for  nearly  twice  as  long. 

A  large  number  of  stars  may  thus  be  observed  on  the  same 
night ;  and  it  will  be  well  to  record  half  of  them  by  the  clock 
at  one  station,  and  the  other  half  by  the  clock  at  the  other 
station,  upon  the  general  principle  of  varying  the  circumstances 
under  which  several  determinations  are  made,  whenever  practi- 
cable, without  a  sacrifice  of  the  integrity  of  the  method.  For 
this  reason,  also,  the  transit  instruments  should  be  reversed 
during  a  night's  work  at  least  once,  an  equal  number  of  stars 
being  observed  in  each  position,  whereby  the  results  will  be 
freed  from  any  undetermined  errors  of  collimation  and  inequality 
of  pivots.  Before  and  after  the  exchange  of  the  star  signals, 
each  observer  should  take  at  least  two  circumpolar  stars  to 
determine  the  instrumental  constants  upon  which  r  and  T' 
depend.  This  part  of  the  work  must  be  carried  out  with  the 
greatest  precision,  employing  only  standard  stars,  as  the  errors 
of  T  and  r'  come  directly  into  the  difference  of  longitude.  The 
right  ascensions  of  the  "signal  stars"  do  not  enter  into  the 
computation,  and  the  result  is,  therefore,  wholly  free  from  any 
error  in  their  tabular  places :  hence  any  of  the  stars  of  the 
larger  catalogues  may  be  used  as  signal  stars,  and  it  will  always 
be  possible  to  select  a  sufficient  number  which  culminate  at 
moderate  zenith  distances  at  both  stations,  (unless  the  difference 
of  latitude  is  unusually  great),  so  that  instrumental  errors  will 
have  the  minimum  effect. 

A  single  night's  work,  however,  is  not  to  be  regarded  as  con- 
clusive, although  a  large  number  of  stars  may  have  been  ob- 
served and  the  results  appear  very  accordant;  for  experience 
show^s  that  there  are  always  errors  which  are  constant,  or  nearly 
so,  for  the  same  night,  and  which  do  not  appear  to  be  represented 
in  the  corrections  computed  and  applied.  Their  existence  is 
proved  when  the  mean  results  of  different  nights  are  compared. 
Moreover,  it  is  necessary  to  interchange  the  observers  in  order 
to  eliminate  their  personal  equations.  The  rule  of  the  Coast 
Survey  has  been  that  when  fifty  stars  have  been  exchanged  on 
not  less  than  three  nights,  the  observers  exchange  stations,  and 
fifty  stars  are  again  exchanged  on  not  less  than  three  nights. 
The  observers  should  also  meet  and  determine  their  relative 
personal  equation,  if  possible,  before  and  after  each  series,  as  it 
may  prove  that  this  equation  is  not  absolutely  constant. 


S46 


LONGITUDE. 


Before  entering  upon  a  series  of  star  signals,  each  observer 
will  be  provided  with  a  list  of  the  stars  to  be  employed.  The 
preparation  of  this  list  requires  a  knowledge  of  the  approximate 
difference  of  longitude  in  order  that  the  stars  may  be  so  selected 
that  transits  at  the  two  stations  may  not  occur  simultaneously. 

EXAMPLE. — For  the  purpose  of  finding  the  difference  of  longi- 
tude between  the  Seaton  Station  of  the  TJ.  S.  Coast  Survey  and 
Raleigh,  a  list  of  stars  was  prepared,  from  which  I  extract  the 
following  for  illustration.  The  latitudes  are 

Seaton  Station  (Washington)        y>  =  -f-  38°  53'.4 
Raleigh      "        (North  Carolina)  ^  =  -f  35    47  .0 

and  Raleigh  is  assumed  to  be^west  from  Washington  6OT  30*. 


Star. 

Mag. 

a 

6 

Seaton  sidereal 
time  of  Raleigh 
transit. 

No.  5036  B.A.C. 

3 

15*    9-  36' 

+  33°  52' 

15*  16"    6' 

5084 

4.3 

18    58 

37    54 

25    28 

5131 

4* 

27      2 

31    51 

33    32 

5192 

5 

36    35 

26    46 

43      5 

5259 

5 

45    43 

36      7 

52    13 

5322 

4i 

55    59 

23    12 

16      2    29 

5388 

5 

16      4      9 

45    19 

10    39 

5463 

3.4 

15    21 

46    40 

21    51 

The  following  table  contains  the  observations  made  on  one  of 
these  stars  at  the  above-named  stations  by  the  U.  S.  Coast  Survey 
telegraphic  party  in  1853,  April  28,  under  the  direction  of  Dr. 
B.  A.  GOULD. 

In  this  table  "  Lamp  W."  expresses  the  position  of  the  rotation 
axes  of  the  transit  instruments.  The  1st  column  contains  the  sym- 
bols by  which  the  fifteen  threads  of  the  three  middle  tallies  were 
denoted;  the  2d  column,  the  times  of  transit  of  the  star  over 
each  thread  at  Seaton,  as  read  from  the  chronographs  at  Seaton ; 
the  3d  column,  the  times  of  these  transits  as  read  from  the  chro- 
nographs at  Raleigh  ;  the -4th  column,  the  mean  of  the  2d  and  3d 
columns ;  the  5th  column,  the  reduction  of  each  thread  to  the 
mean  of  all,  computed  from  the  known  equatorial  intervals  of 
the  threads ;  the  6th  column,  the  time  of  the  star's  transit  over 


BY   THE    ELECTRIC    TELEGRAPH. 


347 


the  mean  of  the  threads,  being  the  algebraic  sum  of  the  numbers 
in  the  4th  and  5th  columns ;  and  the  remaining  columns,  the 
Raleigh  observations  similarly  recorded  and  reduced. 


SEATON—  RALEIGH,  1853  April  28.                              Star  No.  5259  B.  A.  C. 

Seaton  Obs.    Lamp  W. 

Raleigh  Obs.    Lamp  W. 

Thread. 

T, 

R 

Mean. 

Red. 

Tt+T, 

* 

* 

Mean 

Red. 

r/+? 

2 

2 

c, 

§ 

1 

| 

37».97 
41  .37 
44  .03 
47  .81 
50.76 

56.96 
0.06 
15*  46»  3  .40 
6.70 
9.58 

16.03 
19  .26 
22.47 

25  .60 
28.60 

38«.00 
41  .34 
44  .21 
47  74 
50.70 

57  .10 
0  .04 
3  .38 
6.70 
9.58 

15  .93 
19  .30 
22  .45 
25  .60 
28.70 

37»98 
41  .36 
44  .12 
47  .78 
50.73 

57  .03 
0  .05 
3.39 
6.70 
9.58 

15  .98 
13  .28 
.J2.46 
25.60 
23  .65 

+  25«.49 
22  .21 
19  .06 
15  .71 
12.71 

6.21 
3  .25 
+    0  .05 
—    3  .03 
6.28 

12  .54 
15  .83 
18  .09 

22  .23 
25  .33 

Mean 

3«.47 
3.57 
3.18 
3.49 
3.44 

3,24 
3.30 
3.44 

[3  .67] 
3.30 

3.44 
3.45 
3.47 

3.38 
3.32 

17  '.60 
20  .88 
23.90 

30.19 
33.34 

|15»  52"»  36  .40 
39.61 
43.00 

49.04 
52  .30 
65  .50 
b8.73 
2.08 

ll'.OO 
14.50 
17  .55 
20  .79 
23.87 

30.05 
33  .25 
36.30 
39  .53 
43  .00 

48  .81 

52.33 
55  .41 

58  .60 
2.08 

ll'.OO 
14  .54 
17  .58 
20  .84 
23.89 

30  .12 
33  .30 
36  .35 
39  .57 
43  .00 

48.92 
52  .32 
55  .46 

58  .67 
2.08 

22  !25 
19  ,(i5 
15.85 
12.70 

6.32 
3.18 
+    0  .07 
—    3.16 
6.36 

12.75 
15  .90 
19.10 

22.20 
25.38 

Mean  = 

36».45 
36.79 
36  .(S3 
36.69 
36.59 

36.44 
36.48 
36  .42 
36.41 
36.64 

[36  .17] 
36  .42 
36  .36 
36  .47 
36.70 

=  3  .392  l 

=  36  .535 

The  numbers  in  the  last  column  for  each  station  would  be  equal 
if  the  observations  and  chronographic  apparatus  were  perfect ; 
and  by  carrying  them  out  thus  individually  we  can  estimate  their 
accuracy.  The  numbers  [3.67]  at  Seaton  and  [36.17]  at  Raleigh 
are  rejected  by  the  application  of  PEIRCE'S  Criterion  (see  Ap- 
pendix, Method  of  Least  Squares),  and  the  given  means  are 
found  from  the  remaining  numbers. 

The  corrections  of  the  transit  instruments  for  this  star 
(d  =  -f  36°  6'.9)  were 

for  the  Seaton  instrument,  r  =  —  0'.028 
"     "     Raleigh          "  r' =  —  0.193 

The  rate  of  the  clock  was  insensible  in  the  brief  interval 
T7/ —  T.  Hence,  neglecting  the  personal  equations  of  the  ob- 
servers, the  difference  of  longitude  is  found  as  follows : 

i  (T/+  2V)  -f  ?'==  15*  52-  368.342 

i  (T7,  -f  Ta  )  -f  r  =  15  46      3.364 

^=          6    32.978 

In  this  manner  seven  other  stars  were  observed  on  the  same 
night,  and  the  results  were  as  follows : 


348 


LONGITUDE. 


Star 

i 

Diff.  from  mean 

5036  B.  A.  C. 

6-  33«.03 

+  08.04 

5084        « 

33  .09 

-f  0.10 

5131         " 

32.91 

—  0.08 

5192        " 

33.00 

4-  o.oi 

5259 

32.98 

-0.01 

5322 

33  .00 

-f  0.01 

5388 

33.02 

+  0.03 

5463 

32.91 

—  0.08 

Mean  xt  =  6  32  .99 

From  the  residuals  v,  we  deduce  the  mean  error  of  a   single 
determination  by  one  star, 


\\     7 
and  hence  the  mean  error  of  the  value  6"1  32*.  99  is 


0«.Q6 

V8 


=  -+-  0-.02 


But  this  error  will  be  somewhat  increased  by  those  errors  of  the 
instruments  which  are  constant  for  the  night,  and  not  represented 
in  r  and  r',  and  by  the  errors  of  the  personal  equations  yet  to  be 
applied.  Moreover,  a  greater  number  of  determinations  should 
be  compared,  in  order  to  arrive  at  a  just  evaluation  of  the  mean 
error. 

- 
228.  Velocity  of  the  galvanic  current.  —  Recurring  to  the  equations 

of  p.  343,  we  find,  by  taking  the  difference  between  the  values 
of  ),  given  by  the  chronographic  records  at  the  two  stations, 


If  the  clock  is  at  the  eastern  station  (A),  the  time  T2  will  not 
differ  from  T^  except  in  consequence  of  irregularities  in  the 
chronographs  and  errors  in  reading  them,  and  therefore  we 
should  find  x  solely  from  the  times  71/  and  Tz',  or 


(405) 


BY  THE  ELECTRIC  TELEGRAPH.  3-19 

In  like  manner,  if  the  clock  is  at  the  western  station,  we  find  X 
by  the  formula 


Thus,  in  general,  the  transmission  time  will  be  deduced  by  com- 
paring the  records  of  the  star  signals  made  at  one  station  when 
the  clock  is  at  the  other  station. 

In  the  above  example,  the  clock  was  at  Washington,  and 
hence,  from  the  record  of  the  transit  at  Raleigh,  we  have  fourteen 
values  of  71/—  TJ=2x9  as  follows: 

-f-  O.08  -f  O'.OS 

+  .05  -f-  00 

-f-  .09  -f-  .23 

-f  .03  —  .03 

-f  .14  +  .09 

+  .09  -f  .13 

-f  .10  +  .00 

That  these  are  not  merely  accidental  residuals  is  shown  by 
the  permanence  of  sign,  with  the  single  exception  in  the  case 
of  the  eleventh  observation.  The  discrepancies  between  them 
indicate  accidental  variations  in  the  chronographs,  combined  with 
errors  in  reading  off  the  record.  Taking  the  mean,  as  elimi- 
nating to  a  certain  extent  these  errors,  we  have 

2x  =  0*.077  x  =-  0-.0385 

From  this  value  of  x  and  the  distance  of  the  stations  we  can 
deduce  the  velocity  per  second  of  the  galvanic  current.  In  the 
present  instance,  the  length  of  the  wire  was  very  nearly  300 
miles,  and,  if  the  above  single  observation  could  be  depended 

upon,  we  should  have,  velocity  per  second  —  —    —  —  =  7792  miles, 

U.UooO 

which  is  doubtless  too  small. 

The  velocity  thus  found,  however,  appears  to  depend  upon 
the  intensity  of  the  current,*  as  has  been  shown  by  varying  the 
battery  power  on  different  nights.  It  has  also  been  found  that 
the  velocities  determined  from  signals  made  at  the  east  and  west 
stations  differed,  and  that  this  difference  was  apparently  depend- 

*  It  depends  also  upon  tho  sectional  area,  molecular  structure,  and,  of  course, 
material,  of  the  vrire. 


350  LONGITUDE. 

eiit  upon  the  strength,  of  the  batteries;  the  velocities  from  signals 
east-west  and  signals  west-east  coming  out  more  and  more 
nearly  equal  as  the  strength  of  the  batteries  was  increased.  See 
Dr.  GOULD'S  Report  on  telegraphic  determinations  of  differ- 
ences of  longitude,  in  the  Report  of  the  Superintendent  of  the 
U.  S.  Coast  Survey  for  1857,  Appendix  No.  27. 


FOURTH    METHOD. — BY    MOON    CULMINATIONS. 

229.  The  moon's  motion  in  right  ascension  is  so  rapid  that 
the  change  in   this  element  while  the  moon    is   passing   from 
one  meridian  to  another  may  be  used  to  determine  the  difference 
of  longitude.     Its  right  ascension  at  the  instant  of  its  meridian 
transit  is  most  accurately  found  by  means  of  the  interval  of 
sidereal  time  between  this  transit  and  that  of  a  neighboring  well- 
known  star.     For  this  purpose,  therefore,  the  Ephemerides  con- 
tain a  list  of  moon-culminating  stars,  which  are  selected  for  each 
day  so  that  at  least  four  of  them  are  given,  the  mean  of  whose 
declinations  is  nearly  the  same  as  that  of  the  moon  on  that  day, 
and,  generally,  so  that  two  precede  and  two  follow  the  moon. 
The  Ephemerides  also  contain  the  right  ascension  of  the  moon's 
bright   limb  for   each  culmination,  both  upper  and  lower,  and 
the  variation  of  this  right  ascension  in  one  hour  of  longitude, 
— i.e.   the  variation   during   the   interval   between   the    moon's 
transits  over  two  meridians  whose  difference  of  longitude  is  one 
hour.     This  variation  is  not  uniform,  and  its  value  is  given  for 
the  instant  of  the  passage  over  the  meridian  of  the  Ephemeris. 
These  quantities  facilitate  the  reduction  of  corresponding  obser- 
vations, as  will  be  seen  below. 

230.  As  to  the  observation,  let 

#,  #'  =  the  sidereal  times  of  the  culmination  of  the  moon's 
limb  and  the  star,  respectively,  corrected  for  all  the 
known  errors  of  the  transit  instrument,  and  for  clock 
rate, 

a,  a'  =  the  right  ascensions  of  the  moon's  limb  and  the  star 
at  the  instants  of  transit; 

then  we  evidently  have 

o  =  a'  -j-  ft  —  #'  (406) 


BY    MOON    CULMINATIONS.  351 

The  star  and  the  moon  being  nearly  in  the  same  parallel,  the 
instrumental  errors  which  affect  #  also  affect  &'  by  nearly  the 
same  quantity.  We  should  not,  however,  for  this  reason  omit 
to  apply  all  the  corrections  for  known  instrumental  errors,  since 
by  this  omission  we  should  introduce  an  error  in  the  longitude 
precisely  equal  to  the  uncorrected  error  of  the  instrument.  For 
if  the  instrumental  error  produces  the  error  z  in  the  time  of  the 
star's  transit,  the  effect  is  the  same  as  if  the  instrument  were 
perfectly  mounted  in  a  meridian  whose  longitude  west  of  the 
place  of  observation  is  equal  to  z ;  but  the  sidereal  time  required 
by  the  moon  to  describe  this  interval  z  is  equal  to  z  -f-  the 
increase  of  the  moon's  right  ascension  in  this  interval.  Hence 
the  longitude  found,  by  the  methods  hereafter  given,  would  be 
in.  error  by  the  quantity  z. 

231.  If  the  lunar  tables  were  perfectly  accurate,  the  true 
longitude  given  by  the  observation  would  be  found  at  once  by 
comparing  the  observed  right  ascension  with  that  of  the  Ephe- 
meris.  There  are  two  methods  of  avoiding  or  eliminating  the 
errors  of  the  Ephemeris.  In  the  first,  which  has  heretofore  been 
exclusively  followed,  the  observation  is  compared  with  a  corre- 
sponding one  on  the  same  day  at  the  first  meridian,  or  at  some 
meridian  the  longitude  of  which  is  well  established.  In  this 
method  the  increase  of  the  right  ascension  in  passing  from  one 
meridian  to  the  other  is  directly  observed,  and  the  error  of  the 
Ephemeris  on  the  day  of  observation  is  consequently  avoided ; 
but  observations  at  the  unknown  meridian  are  frequently  ren- 
dered useless  by  a  failure  to  obtain  the  corresponding  observa- 
tion at  the  first  meridian. 

In  the  second  method,  proposed  by  Professor  PEIRCE,  the 
Ephemeris  is  first  corrected  by  means  of  all  the  observations 
taken  at  the  fixed  observatories  during  the  semi-lunation  within 
which  the  observation  for  longitude  falls.  The  corrected  Ephe- 
meris then  takes  the  place  of  the  corresponding  observation,  and 
is  even  better  than  the  single  corresponding  observation,  since 
it  has  been  corrected  by  means  of  all  the  observations  at  the 
fixed  observatories  during  the  semi-lunation. 

I  shall  consider  first  the  method  of  reducing  corresponding 
observations. 


352 


LONGITUDE. 


282.  Corresponding  observations  at  places  whose  difference  of  longi- 
tude is  less  than  two  hours. — At  each  place  the  true  sidereal  times 
of  transit  of  the  moon-culminating  stars  and  of  the  moon's 
bright  limb  are  to  be  obtained  with  all  possible  precision :  from 
these,  according  to  the  formula  (406),  will  follow  the  right  as- 
cension of  the  moon's  limb  at  the  instants  of  transit  over  the 
two  meridians,  taking  in  each  case  the  mean  value  found  from 
all  the  stars  observed.  Put 

Lr  Lz  =  the  approximate  or  assumed  longitudes, 

>l  —  the  true  difference  of  longitude, 
a1}  a2  =  the  observed  right  ascensions  of  the  moon's  bright 

limb  at  Ll  and  L2  respectively, 
HQ  =  the  variation  of  the  R.  A.  of  the  moon's  limb  for 


lh  of  longitude  while  passing  from  L^  to 


then  we  have 


(407) 


in  which,  a2  —  at  and  H0  being  both  expressed  in  seconds,  X  will 
be  in  hours  and  decimal  parts. 

When  the  difference  of  longitude  is  less  than  two  hours,  it 
is  found  to  be  sufficiently  accurate  to  regard  HQ  as  constant, 
provided  we  employ  its  value  for  the  middle  longitude 
LQ  =  J(-^i  ~i~  -A>)>  found  by  interpolation  from  the  values  in  the 
Ephemeris,  having  regard  to  second  differences. 

EXAMPLE. — The  following  observations  were  made,  May  15, 
1851,  at  Santiago,  Chili,  by  the  U.  S.  Astronomical  Expedition 
under  Lieut.  GILLISS,  and  at  Philadelphia,  by  Prof.  KENDALL  : 


Object. 

Santiago  sicl.  time. 

Philad'a  sid.  time. 

#  Librae 
Moon  II  Limb 
B.  A.  C.  5579 

15*40-"    3*.37 
16   21    36.84 
16   33    40.12 

15A  45-  228.33 
16   21    39.11 
16   32    58.96 

We  shall  assume  the  longitudes  from  Greenwich  to  be, 

Philadelphia,  L,  =  5*    0-  39'.85 
Santiago,        £3  =  4   42    19. 

the  longitude  of  Philadelphia  being  that  which  results  from  the 
last  chronometric  expeditions  of  the  U.  S.  Coast  Survey,  and 
that  of  Santiago  the  value  which  Lieut.  GILLISS  at  first  assumed. 


BY   MOON    CULMINATIONS. 


353 


The  apparent  right  ascensions  of  the  stars  on  May  15,  by  the 
moon-culminating  list  in  the  Nautical  Almanac,  were 


#  Librae 

B.  A.  C.  5579 


15*  45™  22«.59 
16   32    59.20 


We  have  then  at  Philadelphia,  hy  (406), 


-f  36-  16-.78 
-11    19.85 


16*  21™  39«.37 
16   21    39.35 


Mean  Ol  =  16   21    39  .36 


$  Librae 

B.  A.  C.  5579 


and  at  Santiago : 

#  Librae 

B.  A.  C.  5579 


Hence 


We  shall  find   HQ  for  the   mean   longitude   L0  —  J  (Z^  -f 
=  4*.  86,  by  the  interpolation  formula  (72),  or 


+  35    33  .47 

-  12      3  .28 


16   20    56.06 
16   20    55.92 


Mean  <*,  =  16   20    55  .99 
a3  —  04  =  —  43*.37 


in  which,  if  we  put  n  = 
A  =  n  =  0.405 


4*.86 
12 


we  have 


=     _ 


and  a'  and  60  are  found  from  the  values  of  H  in  the  Ephemeris 
as  follows : 


May  15,  L.  C. 
«     15,  U.  C. 


142-.56 
143  .48 


—  Os.28 


+  0.64    [-0.35] 


«     16,  L.  C. 
"     16,  U.  C. 


4'12+0.23 
144.35  ^ 


whence 

H  =  143-.48          «'==  0-.64  bQ  =  £  (—  (K28  —  0-.41)  =  —  0-.35 

H0  =  143'.48  -f  0*.259  -f  0'.042  =  143«.781 


143.781 


=  —  0\30164  =  — 


5-.90 


354  LONGITUDE. 

which  is  the  longitude  of  Santiago  from  Philadelphia.  Hence, 
if  the  longitude  of  Philadelphia  is  correct,  we  have 

Long,  of  Santiago  =  4*  42m  33«.95  from  Greenwich. 

233.  Corresponding  observations  at  places  whose  difference  of  longi- 
tude is  greater  than  two  hours. — Having  found  at  and  a2  as  in  the 
preceding  case,  we  employ  in  this  case  an  indirect  method  of 
solution.  For  each  assumed  longitude  we  interpolate  the  right 
ascension  of  the  moon's  limb  from  the  Moon  Culminations  in 
the  Ephemeris  to  fourth  differences.  Let 

Av  A2  =  the  interpolated  right  ascensions  of  the  moon's 
limb  for  the  assumed  longitudes  L^  and  L2  respect- 
ively, 

If  the  correction  of  the  Ephemeris  on  the  given  day  is  e, 
the  true  values  of  the  right  ascension  for  TJV  and  L2  are  Al-\-  e 
and  A2  +  e,  the  error  of  the  Ephemeris  being  supposed  to  be 
sensibly  constant  for  a  few  hours ;  but  their  difference  is 

(Az  +  e)-(Al  +  e}  =  A,-Al 

so  that  the  computed  difference  of  right  ascension  is  the  same 
as  if  the  Ephemeris  were  correct.  If  now  the  observed  differ- 
ence a2  —  at  is  the  same  as  this  computed  difference,  the  as- 
sumed difference  of  longitude,  or  L2  —  Lv  is  correct  ;*  but,  if 
this  is  not  the  case,  put 

r^(a3-ai)-(Aa-4)  (408) 

and 

A.L  =  the  correction  of  the  uncertain  longitude,  which  we 
will  suppose  to  be  L^ 

then  f  is  the  change  of  the  right  ascension  while  the  moon  is 
describing  the  small  arc  of  longitude  &L ;  and  for  this  small 
difference  we  may  apply  the  solution  of  the  preceding  article, 
so  that  we  have  at  once 

A£  =  -£-  (in  hours)  (409) 

H 

or 

±£  =  rX  *2  (in  seconds)  (409*) 

H 

*  It  should  be  observed,  however,  that  one  of  the  assumed  longitudes  must  be 
nearly  correct,  for  it  is  evident  that  the  same  difference  of  right  ascension  will  not 
exactly  correspond  to  the  same  difference  of  longitude  if  we  increase  or  decrease 
both  longitudes  by  the  same  quantity. 


BY    MOON   CULMINATIONS.  355 

in  which  the  value  of  H  must  be  that  which  belongs  to  the 
uncertain  meridian  JL2,  or,  more  strictly,  H  must  be  taken  for 
the  mean  longitude  between  L2  and  L2  -f-  &L;  but,  as  A.L  is 
generally  very  small,  great  precision  in  H  is  here  superfluous. 
However,  if  in  any  case  &L  is  large,  we  can  first  find  H  for  the 
meridian  Z/2,  and  with  this  value  an  approximate  value  of  A£; 
then,  interpolating  IT  for  the  meridian  L2-\-%  A£,  a  more  correct 
value  of  &L  will  be  found.* 

EXAMPLE. — The  following  observations  were  made  May  15, 
1851,  at  Santiago  and  Greenwich : 


Object. 

$  Librae 
.  Moon  II  Limb 
B.A.C.  5579 

Santiago. 
15*46™    3'.37 
16   21    36.84 
16  33    40.12 

Greenwich. 
15*  45-  22'.37 
16     9    39.41 
16   32    59.17 

We  assume  here,  as  in  the  preceding  example,  for  Santiago 
Z/2  =  4k  42m  19s,  and  for  Greenwich  we  have  Z/x  =  0.  The  places 
of  the  stars  being  as  in  the  preceding  article,  we  find  for 

Greenwich,  ttl  =  16*    9m  39'.54 

Santiago,      a3  =  16   20    55  .99 

a2  —  a,  =        11    16  .45 

The  computed  right  ascension  for  Greenwich  is  in  this  case 
simply  that  given  in  the  Ephemeris  for  May  15 ;  the  increase  to 
the  meridian  4A  42™  19*. 0  has  been  found  in  our  example  of  in- 
terpolation, Art.  71,  to  be 

4,— 4  =  ll«15-.84 
and  hence 

r  =  +  0-.61 

We  find,  moreover,  for  the  longitude  4fc  42™  19*, 

Zf=143'.77 

whence 


By  these  observations  we  have,  therefore, 

Longitude  of  Santiago  —  4*  42W  34s. 28 


*  This  method  of  reducing  moon  culminations  was  developed  by  WALKER,  Frant- 
actions  of  the  American  Philosophical  Society,  new  series,  Vol.  V. 


356  LONGITUDE. 

234.  Reduction  of  moon  culminations  by  the  hourly  Ephemeris. — 
The  method  of  reduction  given  in  the  preceding  article  is  per- 
fectly exact ;  but  the  interpolation  of  the  moon's  place  to  fourth 
differences  is  laborious.  The  hourly  Ephemeris,  however,  requires 
the  use  of  second  differences  only.  The  sidereal  time  of  the 
transit  of  the  moon's  centre  at  the  meridian  ^  is  =  the  observed 
right  ascension  of  the  centre  =  <zr  If  then  we  put 

T,  =  the  mean  Greenwich  time  corresponding  to  a,  as  found 

by  the  hourly  Ephemeris, 
0j  =  the  Greenwich  sidereal  time  corresponding  to  Tlf 

we  have  at  once,  if  the  Ephemeris  is  correct, 

A  =  0,  -  04  (410) 

This,  indeed,  was  one  of  the  earliest  methods  proposed,  but  was 
abandoned  on  account  of  the  imperfection  of  the  Ephemeris. 
The  substitution  of  corresponding  observations,  however,  does 
not  require  a  departure  from  this  simple  process ;  for  we  shall 
have  in  the  same  manner,  from  the  observations  made  at 
another  meridian  (which  may  be  the  meridian  of  the  Ephemeris), 

i3  =  ea-0a 

and  hence 

I  =  A  -  £,  =  (0,    -  e.)  -  (a,  -  «,)  (411) 

and  it  is  evident  that  the  difference  (0t  —  00)  of  the  Greenwich 
times  will  be  correct,  although  the  absolute  right  ascension  of 
the  Ephemeris  is  in  error,  provided  the  hourly  motion  is  correct. 
The  correctness  of  the  hourly  motion  must  be  assumed  in  all 
methods  of  reducing  moon  culminations ;  and  in  the  present 
state  of  the  lunar  theory  there  can  be  no  error  in  it  which  can 
be  sensible  in  the  time  required  by  the  moon  to  pags  from  one 
meridian  to  another. 

In  this  method  a  is  the  right  ascension  of  the  moon's  centre 
at  the  instant  of  the  transit  of  the  centre ;  which  may  be  de- 
duced from  the  time  of  transit  of  the  limb  by  adding  or  sub- 
tracting the  "  sidereal  time  of  semidiameter  passing  the  meridian," 
given  in  the  table  of  moon  culminations  in  the  Ephemeris.* 

To  find  Ti  corresponding  to  av  we  may  proceed  as  in  Art.  64, 

_» 

*  If  we  wish  to  be  altogether  independent  of  the  moon-culminating  table,  we  can 
compute  the  sidereal  time  of  semidiameter  passing  the  meridian  by  the  formula  (see 
Vol.  II.,  Transit  Instrument), 


BY    MOON    CULMINATIONS.  357 

or  as  follows:     Let  T^  and  T0-\-  1*  be  the.  two  Greenwich  hours 
.between  which  at  falls,  and  put 

Aa  =  the  increase  of  right  ascension  in  lm  of  mean  time  at 

the  time  T0, 

da  =  the  increase  of  Aa  in  1*, 
a0  =  the  right  ascension  of  the  Ephemeris  at  the  hour  T9, 

then,  by  the  method  of  interpolation  by  second  differences,  we 
have 


3600  60 


in  which  the  interval  T^  —  TQ  is  supposed  to  be   expressed  in 
seconds.     This  gives 


—  T  = 


60(04-0 


i         -o  XT          T 

da     ll  —  1Q 


~  2       3600 

and  in  the  second  member  an  approximate  value  of  Tv  may  be 
used,  deduced  from  the  local  time  of  the  observation  and  an 
approximate  longitude.  A  still  more  convenient  form,  which 
dispenses  with  finding  an  approximate  value  of  Tv  is  obtained 
as  follows :  Put 

then  we  have 


S 


15(1  —  A)cosf> 

in  which  S  —  the  moon's  semidiameter,  A  =  the  increase  of  the  moon's  right  ascen- 
sion in  one  sidereal  second,  and  6  =  the  moon's  declination,  which  are  to  be  taken 
for  the  Greenwich  time  of  the  observation,  approximately  known  from  the  local  time 
and  the  approximate  longitude. 

Or  we  may  apply  to  the  sidereal  time  (—  tfj)  of  the  transit  of  the  limb  the  quantity 


15  cosd 

and  the  resulting  a:=r  ^  ±  ^  S  sec  6  will  be  the  right  ascension  of  the  moon's 
centre  at  the  local  sidereal  time  -&r  We  then  find  the  Greenwich  time  Ql  corre- 
sponding to  aj  as  in  the  text,  and  we  have 


358  LONGITUDE. 


„_  6Q(a.-a0) 

X  ~         ~^  X 


__ 

r  7200 


ov,  with  sufficient  accuracy, 

/ 


7200   A 
Putting  then 

T,=  so  (,.-»,)      1,,==_5^.A  (412) 

Aa  7200    Aa 

we  have,  very  nearly, 

x^x'+.x"  (413) 

As  a  practical  rule  for  the  computer,  we  may  observe  that  x" 
will  be  a  positive  quantity  when  AOC  is  decreasing,  and  negative 
when  AO,  is  increasing. 

The  method  of  this  article  will  be  found  particularly  conve- 
nient when  the  observation  is  compared  directly  with  the 
Ephemeris,  the  latter  being  corrected  by  the  following  process. 
See  page  362. 

235.  Peirce's  method  of  correcting  the  Ephemeris*  —  The  accuracy 
of  the  longitude  found  by  a  moon  culmination  depends  upon 
that  of  the  observed  difference  of  right  ascension.  When  this 
difference  is  obtained  from  two  corresponding  observations,  if 
the  probable  errors  of  the  observed  right  ascensions  at  the  two 
meridians  are  e,  and  e2,  the  probable  error  of  the  difference  will 
be  =  v/^2  -f  e22).  [Appendix].  But  if  instead  of  an  actual  ob- 
servation at  _L2  we  had  a  perfect  Ephemeris,  01^  £2=  0,  the 
probable  error  of  the  observed  difference  would  be  reduced  to  et; 
and  if  we  have  an  Ephemeris  the  probable  error  of  which  is  less 
than  that  of  an  observation,  the  error  of  the  observed  difference 
is  reduced.  At  the  same  time,  we  shall  gain  the  additional 
advantage  that  every  observation  taken  at  the  meridian  whose 
longitude  is  required  will  become  available,  even  when  no  corre- 
sponding observation  has  been  taken  on  the  same  day;  and 

*  Report  of  the  Superintendent  of  the  U.  S  Coast  Survey  for  1854,  Appendix, 
p.  115*. 


BY    MOON    CULMINATIONS.  359 

experience  has  shown  that,  when  we  depend  on  corresponding 
observations  alone,  about  one-third  of  the  observations  are 
lost. 

The  defects  of  the  lunar  theory,  according  to  PEIRCE,  are 
involved  in  several  terms  which  for  each  lunation  may  be 
principally  combined  into  two,  of  which  one  is  constant  and  the 
other  has  a  period  of  about  half  a  lunation,  and  he  finds  that 
for  all  practical  purposes  we  may  put  the  correction  of  the 
Ephemeris  for  each  semi-lunation  under  the  form 

X=  A  -f  Bt+  CP  (414) 

in  which  A,  -B,  and  C  are  constants  to  be  determined  from  the 
observations  made  at  the  principal  observatories  during  the 
semi-lunation,  and  t  denotes  the  time  reckoned  from  any  assumed 
epoch,  which  it  will  be  convenient  to  take  near  the  mean  of  the 
observations.  The  value  of  t  is  expressed  in  days ;  and  small 
fractions  of  a  day  may  be  neglected.  Let 

al7  a3,  as,  &c.  =  the  right  ascension  observed  at  any  observa- 
tory at  the  dates  tv  ta,  ta)  &c.,  from  the  assumed 
epoch, 

Oj',  aa',a3',&c.  =  the  right  ascension  at  the  same  instant  found 
from  the  Ephemeris, 

and  put 


then  nv  nz,  n3,  &c.  are  the  corrections  which  (according  to  the 
observations)  the  Ephemeris  requires  on  the  given  dates,  and 
hence  we  have  the  equations  of  condition 

A  -f  Stt  +  Ct*  —  ^  =  0 
A  +  Bta  +  Ct*  —  n2  =  0 
A  +  Btt  +  CV  —  n3  =  0 

&c. 
f 

In  order  to  eliminate  constant  errors  peculiar  to  any  observa- 
tory, when  the  observation  is  not  made  at  Greenwich,  the  ob- 
served right  ascension  is  to  be  increased  by  the  average  excess 
for  the  year  (determined  by  simultaneous  observations)  of  the 
right  ascensions  of  the  moon's  limb  made  at  Greenwich  above 
those  made  at  the  actual  place  of  observation. 


360 


LONGITUDE. 


If  now  we  put 

m  =  the  number  of  observations  =  the  number  of  equations 

of  condition, 

T  =  the  algebraic  sum  of  the  values  off, 
Ta  =  the  sum  of  the  squares  of  t , 
T3=  the  algebraic  sum  of  the  third  powers  off, 
Tt—  the  sum  of  the  fourth  powers  off, 
jV  =  the  algebraic  sum  of  the  values  of  n, 
JWl=  the  algebraic  sum  of  the  products  of  n  multiplied  by  f, 
Nz=  the  algebraic  sum  of  the  products  of  n  multiplied  by  f2, 

the  normal  equations,  according  to  the  method  of  least  squares, 
will  be 

mA  4-  TB    4-  TaC  —  N  =  0  ^ 

TA  -f-  T,B  4-  T3C—Nt=0  V  (415) 

T3B+  T,C-N2=0  ) 


The  solution  of  these  equations  by  the  method  of  successive 
substitution,  according  to  the  forms  given  in  the  Appendix,  may 
be  expressed  as  follows  : 


TT 

-*- 


T,  =  T       ^_ 


m 


T"  =  T!— 


TN 

m 

TZ 

m 


T1  AT' 
3     l 


N" 
C  — 


B=- 


,'—  T3'C 

T: 


A  = 


N—  T2C  — 
m 


(416) 


Then,  to  find  the  mean  error  of  the  corrected  Ephemeris,  we 
observe  that  this  error  is  simply  that  of  the  function  X,  which  is 
to  be  found  by  the  method  of  the  Appendix,  according  to  which 
we  first  find  the  coefficients  &0,  kv  k2  by  the  following  formulae  : 


and  then,  putting 


mkQ+ 


BY   MOON   CULMINATIONS. 


361 


we  have 


(e-T)  =  Ms 


(417) 


in  which  e  denotes  the  mean  error  of  a  single  observation  and 
(e^T)  the  mean  error  of  the  corrected  Ephemeris  ;  or,  if  e  denotes 
the  probable  error  of  an  observation,  (eJT)  denotes  the  probable 
error  of  the  corrected  Ephemeris.  (Appendix.) 

If  the  values  of  /t0,  k^  and  k2  are  substituted  in  M  ,  we  shall  have 


It  will  generally  happen,  where  a  sufficient  number  of  observa- 

T' 

tions  are  combined,  that  -~-  is  a  small  fraction  which  may  be 

•*« 

neglected  without  sensibly  affecting  the  estimation  of  a  probable 
error,  and  we  may  then  take 


—  \/l  : 


1       (t  — 


+ 


(418*) 


According  to  PEIRCE,  the  probable  error  of  a  standard  observa- 
tion of  the  moon's  transit  is  (K104  (found  from  the  discussion  of 
a  large  number  of  Greenwich,  Cambridge,  Edinburgh,  and  "Wash- 
ington observations) ;  so  that  the  probate  error  of  the  corrected 
Ephemeris  will  be  equal  to  M.  (OM04). 

EXAMPLE. — At  the  Washington  Observatory,  the  following 
right  ascensions  of  the  moon  were  obtained  from  the  transits  over 
twenty-five  threads,  observed  with  the  electro-chronograph : 


Approx.  Green.  Mean  Time. 

R.  A.  of 
3  II  Limb. 

Sid.  time  semid. 
passing  merid. 

R.  A.  of  })  centre 
=  *r 

1859,  Aug.  16,  19* 
«      17,  20 

"      18,  21 

0*    8-53-.40 
0   54    33.57 
1   42    48.53 

62'.06 
63.54 
65.77 

0*    7-51-.34 
0   53    30.03 
1   41    42.76 

The  sidereal  time  of  the  semidiameter  passing  the  meridian  is 
here  taken  from  the  British  Almanac,  as  we  propose  to  reduce  the 
observations  by  means  of  the  Greenwich  observations  which  are 
reduced  by  this  almanac.  We  thus  avoid  any  error  in  the  semi- 
diameter. 

During   the   semi-lunation   from   Aug.  13   to   Aug.    27,   the 
Greenwich  observations,  also  made  with  the  electro-chronograph, 


362 


LONGITUDE. 


gave  the  following  corrections  (=  n)  of  the  Nautical  Almanac 
right  ascensions  of  the  moon : 


Approx.  Greenwich  Mean  Time. 

n 

t 

1859".  Aug.  14,  13* 

—  0-.39 

—  3. 

"      15,  14 

—  0.26 

—  1.9 

«      16,  14 

—  0.49 

—  0.9 

«      18,  16 

—  0.63 

+  1.2 

«      19,  17 

—  1.04 

1    2.2 

«      20,  17 

—  1.08 

+  3.2 

Let  us  employ  these  observations  to  determine  by  Peirce's 
method  the  most  probable  correction  of  the  Ephemeris  on  the 
dates  of  the  Washington  observations.  Adopting  as  the  epoch 
Aug.  17th  12A  or  17d.5,  the  values  of  t  are  approximately  as  above 
given.  The  correction  of  the  Ephemeris  being  sensibly  constant 
for  at  least  one  hour,  these  values  are  sufficiently  exact.  We 
find  then 

Tr^0.8 

m  =  6 

and  hence,  by  (416), 

C  =  —  08.02135  B  =  •—  0-.1257  A  =  —  08.525 

The  correction  of  the  Ephemeris  for  any  given  date  /,  reckoning 
from  Aug.  17.5,  is,  therefore, 

jf  =  _  0'.525  —  O'.1257<  —  08.02135£2 

•  Consequently,  for  the  dates  of  the  Washington  observations, 
the  correction  and  the  probable  error  (Ms)  of  the  correction, 
found  by  (418)  or  (418*),  are  as  follows: 


T3  =     29.94 

Ta  =    10.556 

T4  =    225.045 

T9'  =    29.83 

TV  =      6.564 

T4'  =      75.644 

T4"  =  74.200 

jy   ^_3..89 

Nt=  —  48.41 

N,  =  —  21-.85 

JV/^  —  3.89 

jya'=--    2.44 

N2"  =  —  18.58 

Aug.  16,  19*        t  =  —  0.7 

17,  20         t  =  +  0.3 

18,  21         t  =  +  1.4 


X=  —  0*.45 
JT=  —  0.56 


Ms  =  0-.05 
Me  =  0  .04 
Me  =  0  .04 


The  longitude  of  the  Washington  Observatory  may  now  be 
found  by  the  hourly  Ephemeris  (after  applying  these  correc- 
tions), by  the  method  of  Art.  234.  Taking  the  observation  of 
Aug.  16,  we  have 


BY    MOON    CULMINATIONS.  363 

Aug.  16,  T0  =  19*,  E.  A.  of  Ephemoris  =  0*  6-  47'.56 

JT=_      -  0.45 

Aa   =  1.8122      da  =  -f  0.0023      a0  =  0    6     47  .11 

at  =  0   7    51.34 
4.23 


ao 


logx'2    6.6554 

log  do,    7.3617 

ur.  co.  log  Aa    9.7418 


log  (a,  -  «0)     1.80774 

ar.  co.  log  Aa     9.74179 

log  60     1.77815 

logo/      3.32768 

tf  =  35™  26'.57  log  xf'  n9.9016 

x"  =  -       0.80 
x    =  35   25  .77 

Hence,  Greenwich  mean  time  =  T0  -f  x  =  19*  35™  25*.77 
Sidereal  time  mean  noon  =    9   37    24  .18 

Correction  for  19*  35-  25'.77  3    13.09 

Greenwich  sidereal  time  =    5    16      3 .04 

Local  sidereal  time  =  04  =    0     7    51  .34 

Longitude  =    5     8    11  .70 

The  observations  of  the  17th  and  18th  being  reduced  in  the 
same  manner,  the  three  results  are 


Probable  error.* 

Weight. 

Aug.  16,     5* 

8"  118.70 

S-.5 

1. 

«     17, 

12.50 

3.1 

1.3 

«     18, 

11.10 

2  .9 

1.5 

Mean  by  weights  =  5   < 

3    11.74 

1.8 

236.  Combination  of  moon  culminations  by  weights. — When  some 
of  the  transits  either  of  the  moon  or  of  the  comparison  stars  are 
incomplete,  one  or  more  of  the  threads  being  lost,  such  observa- 
tions should  evidently  have  less  weight  than  complete  ones,  if 
we  wish  to  combine  them  strictly  according  to  the  theory  of 
probabilities.  Besides,  other  things  being  equal,  a  determina- 
tion of  the  longitude  will  have  more  or  less  weight  according  to 
the  greater  or  less  rapidity  of  the  moon's  motion  in  right  ascen- 
sion. 


For  the  computation  of  the  probable  error  and  weight,  see  the  following  article. 


364  LONGITUDE. 

If  the  weight  of  a  transit  either  of  the  moon  or  a  star  were 
simply  proportional  to  the  number  of  observed  threads,  as  has 
been  assumed  by  those  who  have  heretofore  treated  of  this  sub- 
ject,* the  methods  which  they  have  given,  and  which  are  obvious 
applications  of  the  method  of  least  squares,  would  be  quite  suffi- 
cient. But  the  subject,  strictly  considered,  is  by  no  means  so 
simple. 

Let  us  first  consider  the  formula 

a,  =  a!  -f  ^  —  #' 

or,  rather 

•>==*,+  (a'  -  *') 

in  which  $L  and  $'  are  the  observed  sidereal  times  of  the  transit 
of  the  moon  and  star,  respectively;  a'  is  the  tabular  right  ascen- 
sion of  the  star,  and  at  is  the  deduced  right  ascension  of  the 
moon.  The  probable  error  of  aL  is  composed  of  the  probable 
errors  of  ^  and  of  a'  —  $',  which  belong  respectively  to  the 
moon  and  the  star.  "We  may  here  disregard  the  clock  errors,  as 
well  as  the  unknown  instrumental  errors,  since  they  affect  $x 
and  $'  in  the  same  manner,  very  nearly,  and  are  sensibly  elimi- 
nated in  the  difference  $t  --  $r.  The  probable  error  of  the 
quantity  a'  —  $'  is  composed  of  the  errors  of  a'  and  #'.  The 
probable  error  of  the  tabular  right  ascension  of  the  moon-culmi- 
nating stars  is  not  only  very  small,  but  in  the  case  of  correspond- 
ing observations  is  wholly  eliminated ;  and  even  when  we  use 
a  corrected  Ephemeris  it  will  have  but  little  effect,  since  the  ob- 
served right  ascension  of  the  moon  at  the  principal  observatories 
always  depends  (or  at  least  should  depend)  chiefly  upon  these 
stars.  We  may,  therefore,  consider  the  error  of  a'  —  $'  as  sim- 
ply the  error  of  $'.  We  have  here  to  deal  with  those*  errors  only 
\vhich  do  not  necessarily  affect  &'  and  ^  in  the  same  manner, 
and  of  these  the  chief  and  only  ones  that  need  be  considered 
here  are — 1st,  the  culmination  error  produced  by  the  peculiar  con- 
ditions of  the  atmosphere  at  the  time  of  the  star's  transit,  which 
are  constant,  or  nearly  so,  during  the  transit,  but  are  different 
for  different  stars  and  on  different  days;  and,  2d,  the  accidental 
error  of  observation.  It  is  only  the  latter  which  can  be  diminished 

*  NICOLA  j,  in  the  Astronomische  Nachrichten,  No.  26;  and  S.  C.  WALKER,  Transac- 
tions of  the  American  Philosophical  Society,  Vol.  VI.  p.  253. 


BY    MOON    CULMINATIONS.  365 

by  increasing  the  number  of  threads.  In  Vol.  II.  (Transit  In- 
strument) I  shall  show  that  the  probable  error  of  a  single  deter- 
mination of  the  right  ascension  of  an  equatorial  star  (and  this 
may  embrace  the  moon-culminating  stars)  at  the  Greenwich 
Observatory  is  0s. 06,  whereas,  if  the  culmination  error  did  not 
exist  it  would  be  only  0*.03,  the  probable  error  of  a  single 
thread  being  =  0s. 08,  and  the  number  of  threads  =  7.  Hence, 
putting 

c  =  the  probable  culmination  error  for  a  star, 

we  deduce* 

c  =  l/(0.06)2  —  (0.03)2  =  08.052 
If,  then,  we  put 

e  =  the  probable  accidental  error  of  the  transit  of  a  star  over 

a  single  thread, 
n  —  the  number  of  threads  on  which  the  star  is  observed, 

the  probable  error  of  #',  and,  consequently,  also  of  a' —  $',  is 


and  the  weight  of  a'  —  $'  for  each  star  may  be  found  by  the 

formula 

E* 


in  which  E  is  the  probable  error  of  an  observation  of  the  weight 
unity,  which  is,  of  course,  arbitrary.  If  we  make  p  =  1  when 
n  =  7,  we  have  E  =  0*.06.  Substituting  this  value,  and  also 
c  =  0*.052,  s  =  0s.  08,  the  formula  may  be  reduced  to  the  fol- 
lowing : 


P  = 


100  4-  — 

n 

The  value  of  at  is  to  be  deduced  by  adding  to  $t  the  mean 


*  The  value  of  c  thus  found  involves  other  errors  besides  the  culmination  error 
proper,  such  as  unknown  irregularities  of  the  clock  and  transit  instrument,  &c. 
These  cannot  readily.be  separated  from  c,  nor  is  it  necessary  for  our  present  purpose. 


366  LONGITUDE. 

according  to  weights  of  all  the  values  of  al  —  ^  given  by  the 
several  stars,  or 

»'=*<+[-^r°  (420) 

where  the  rectangular  brackets  are  employed  to  express  the  sum 
of  all  the  quantities  of  the  same  form.  The  probable  error  of 
the  last  term  will  be 

E          0«.Q6 


If  now  we  put 

et  —  the  probable  error  of  ax, 
cl  =  the  culmination  error  for  the  moon, 
he   =  the  probable  accidental   error  of  the  transit  of  the 

moon's  limb  over  a  single  thread, 
wx  —  the  number  of  threads  on  which  the  moon  is  observed, 


the  probable  error  of  ^  will  be  —  -v/cl2  +  !cfl,  and  hence 

\  n 


-y++ 

To  determine  cl  I  shall  employ  the  values  of  the  other  quantities 
in  this  equation  which  have  been  found  from  the  Greenwich 
observations.  Professor  PEIRCE  gives  sl=  0SA04^  and  in  the 
cases  which  I  examined  I  found  the  mean  value  k  —  1.3.  As- 
suming [/>]  =  4  as  the  average  number  of  stars  upon  which  a^ 
depends  in  the  Greenwich  series,  we  have 

(0104)'  =  C-+ 

whence 

cl  =  (K091 

and  the  formula  for  the  probable  error  of  al  observed  at  the 
meridian  is 


In  the  case  of  corresponding  observations  at  a  second  meridian 
1/2,  the  probable  error  e2  is  also  to  be  found  by  this  formula,  and 
then  the  probable  error  of  the  deduced  difference  of  right  ascen- 
sion will  be 


BY    MOON    CULMINATIONS.  367 

and  the  probable  error  of  the  deduced  longitude  will  be 


=  hl/el*+£.*  (423) 

where,  H  being  the  increase  of  the  moon's  right  ascension  in  1* 
of  longitude,  we  have 

A  =  ?£!  (424) 

But  if  the  observation  at  the  meridian  L^  is  compared  with  a 
corrected  Ephemeris  (Art.  235)  the  probable  error  of  which  is 
M (0s.  104),  the  probable  error  of  the  deduced  longitude  will  be 


=  h  j/e,2  +  M*  (0.104)2  (425) 

Finally,  all  the  different  values  of  the  longitude  will  be  com- 
bined by  giving  them  weights  reciprocally  proportional  to  the 
squares  of  their  probable  errors. 

The  preponderating  influence  of  the  constant  error  represented 
by  the  first  term  of  (422)  is  such  that  a  very  precise  evaluation 
of  the  other  terms  is  quite  unimportant.  It  is  also  evident  that 
we  shall  add  very  little  to  the  accuracy  of  an  observation  by 
increasing  the  number  of  threads  of  the  reticule  beyond  five  or 
seven.  For  example,  suppose,  as  in  the  "Washington  observations 
used  in  Art.  235,  that  twenty-five  threads  are  taken,  and  that 
four  stars  are  compared  with  the  moon ;  we  have  for  each  star, 

>— »*      i* 


and  hence 


_ 


whereas  for  seven  threads  we  have  £l  —  0*.104,  and  therefore 
the  increase  of  the  number  of  threads  has  not  diminished  the 
probable  error  by  so  much  as  0s.  01. 

For  the  observations  of  1859  August  16,  17,  18,  Art.  235,  the 
values  of  h  are  respectively 

32.1         30.8         and        28.8 

and,  taking  Me  =  M (0*.104)  as  given  in  that  article,  namely, 
0'.05         0*.04         and         0'.04 


368  LONGITUDE. 

with  the  value  of  sl  =  O.097  above  found,  we  deduce  the  proba- 
ble errors  of  the  three  values  of  the  longitude,  by  (425), 

3'.5         38.1         and         2«.9 

The  reciprocals  of  the  squares  of  these  errors  are  very  nearly  in 
the  proportion  of  the  numbers  1,  1.3,  1.5,  which  were  used  as 
the  weights  in  combining  the  three  values. 

237.  The  advantage  of  employing  a  corrected  Ephemeris 
instead  of  corresponding  observations  can  now  be  determined 
by  the  above  equations.  If  the  observations  are  all  standard 
observations  (represented  by  T^  —  7  and  [p]  —  4),  we  shall  have 
sl=  e2=  O.104,  and  the  probable  error  of  the  longitude  will  be 

by  corresponding  observations  =  h^  j/2 


by  the  corrected  Ephemeris       =  Aex  j/1  -j-  M 2 

The  latter  will,  therefore,  be  preferable  when  M  <  1,  which  will 
always  be  the  case  except  when  very  few  observations  have  been 
taken  at  the  principal  observatories. 

But  experience  has  shown  that  when  we  depend  wholly  on 
corresponding  observations  we  lose  about  one-third  of  the 
observations,  and,  consequently,  the  probable  error  of  the  final 
longitude  from  a  series  of  observations  is  greater  than  it  would 
be  were  all  available  in  the  ratio  of  j/3  :  j/2.  Hence  the  proba- 
ble errors  of  the  final  results  obtained  by  corresponding  observa- 
tions exclusively,  and  by  employing  the  corrected  Ephemeris  by 
which  all  the  observations  are  rendered  available,  are  in  the 
ratio  |/3  :  j/1  +  M2]  and,  the  average  value  of  M  being  about 
0.6*  this  is  as  1  :  0.67. 

If,  however,  on  the  date  of  any  given  observation  at  the  meri- 
dian to  be  determined,  we  can  find  corresponding  observations 
at  two  principal  observatories,  the  probable  error  of  the  longitude 
found  by  comparing  their  mean  with  the  given  observation  will 
be  only  h^  j/1-5,  which  is  so  little  greater  than  the  average  error 
in  the  use  of  the  corrected  Ephemeris,  that  it  will  hardly  be 
worth  while  to  incur  the  labor  attending  the  latter.  If  there 
should  be  three  corresponding  observations,  the  error  will  be 
reduced  to  he^  j/1. 33,  and,  therefore,  less  than  the  average  error 
of  the  corrected  Ephemeris. 


BY   MOON   CULMINATIONS.  369 

The  advantage  of  the  new  method  will,  therefore,  be  felt 
chiefly  in  cases  where  either  no  corresponding  observation,  or 
but  one,  has  been  taken  at  any  of  the  principal  observatories. 

238.  The  mean  value  of  h  is  about  —  27,  and  therefore  a 
probable  error  of  0*.l  in  the  observed  right  ascension,  supposing 
the  Ephemeris  perfect,  will  produce  a  mean  probable  error  of  2*.7 
in  the  longitude.  If  the  probable  error  diminished  without 
limit  in  proportion  to  the  square  root  of  the  number  of  observa- 
tions, as  is  assumed  in  the  theory  of  least  squares,  we  should 
only  have  to  accumulate  observations  to  obtain  a  result  of  any 
given  degree  of  accuracy.  But  all  experience  proves  the  fallacy 
of  this  law  when  it  is  extended  to  minute  errors  which  must 
wholly  escape  the  most  delicate  observation.  The  remarks  of 
Professor  PEIRCE  on  this  point,  in  the  report  above  cited,  are  of 
the  highest  importance.  He  says :  "  If  the  law  of  error  embodied 
in  the  method  of  least  squares  were  the  sole  law  to  which 
human  error  is  subject,  it  would  happen  that  by  a  sufficient 
accumulation  of  observations  any  imagined  degree  of  accuracy 
would  be  attainable  in  the  determination  of  a  constant ;  and  the 
evanescent  influence  of  minute  increments  of  error  would  have 
the  effect  of  exalting  man's  power  of  exact  observation  to  an 
unlimited  extent.  I  believe  that  the  careful  examination  of 
observations  reveals  another  law  of  error,  which  is  involved  in 
the  popular  statement  that  '  man  cannot  measure  what  he  cannot 
see.'  The  small  errors  which  are  beyond  the  limits  of  human 
perception  are  not  distributed  according  to  the  mode  recognized 
by  the  method  of  least  squares,  but  either  with  the  uniformity 
which  is  the  ordinary  characteristic  of  matters  of  chance,  or  more 
frequently  in  some  arbitrary  form  dependent  upon  individual] 
peculiarities, — such,  for  instance,  as  an  habitual  inclination  to  the* 
use  of  certain  numbers.  On  this  account,  it  is  in  vain  to  attempt 
the  comparison  of  the  distribution  of  errors  with  the  law  of  least 
squares  to  too  great  a  degree  of  minuteness ;  and  on  this  account, 
there  is  in  every  species  of  observation  an  ultimate  limit  of  accuracy 
beyond  which  no  mass  of  accumulated  observations  can  ever  penetrate. 
A  wise  observer,  when  he  perceives  that  he  is  approaching  this 
limit,  will  apply  his  powers  to  improving  the  methods,  rather 
than  to  increasing  the  number  of  observations.  This  principle 
will  thus  serve  to  stimulate,  and  not  to  paralyze,  effort ;  and  its 

VOL.  I.— 24 


370  LONGITUDE. 

vivifying  influence  will  prevent  science  from  stagnating  into 
mere  mechanical  drudgery. 

*<  In  approaching  the  ultimate  limit  of  accuracy,  the  probable 
error  ceases  to  diminish  proportionably  to  the  increase  of  the 
number  of  observations,  so  that  the  accuracy  of  the  mean  of 
several  determinations  does  not  surpass  that  of  the  single  deter- 
minations as  much  as  it  should  do  in  conformity  with  the  law  of 
least  squares :  thus  it  appears  that  the  probable  error  of  the 
mean  of  the  determinations  of  the  longitude  of  the  Harvard 
Observatory,  deduced  from  the  moon-culminating  observations 
of  1845,  1846,  and  1847,  is  ls.28  instead  of  being  ls.OO,  to  which 
it  should  have  been  reduced  conformably  to  the  accuracy  of  the 
separate  determinations  of  those  years. 

"  One  of  the  fundamental  principles  of  the  doctrine  of  proba- 
bilities is,  that  the  probability  of  an  hypothesis  is  proportionate 
to  its  agreement  with  observation.  But  any  supposed  computed 
lunar  epoch  may  be  changed  by  several  hundredths  of  a  second 
without  perceptibly  affecting  the  comparison  with  observation, 
provided  the  comparison  is  restricted  within  its  legitimate  limits 
of  tenths  of  a  second.  Observation,  therefore,  gives  no  informa- 
tion which  is  opposed  to  such  a  change." 

The  ultimate  limit  of  accuracy  in  the  determination  of  a 
longitude  by  moon  culminations,  according  to  the  same  distin- 
guished authority,  is  not  less  than  one  second  of  time.  This  limit 
can  probably  be  reached  by  the  observations  of  two  or  three 
years,  if  all  the  possible  ones  are  taken;  and  a  longer  continuance 
•of  them  would  be  a  waste  of  time  and  labor. 

From  these  considerations  it  follows  that  the  method  of  moon 
culminations,  when  the  transits  of  the  limb  are  employed,  cannot 
•come  into  competition  with  the  methods  by  chronometers  and 
•occupations  where  the  latter  are  practicable.* 

*  In  consequence  of  the  uncertainty  attending  the  observation  of  the  transit  of 
the  moon's  limb,  it  has  been  proposed  by  MAEDLER  (Astron.  Nach.  No.  337)  to  sub- 
stitute the  transit  of  a  well-defined  lunar  spot.  The  only  attempt  to  carry  out  this 
suggestion,  I  think,  is  that  of  the  U.  S.  Coast  Survey,  a  report  upon  which  by  Mr. 
PETERS  will  be  found  in  the  Report  of  the  Superintendent  for  1856,  p.  198.  The 
varying  character  of  a  spot  as  seen  in  telescopes  of  different  powers  presents,  it 
seems  to  me,  a  very  formidable  obstacle  to  the  successful  application  of  this 
method. 


BY   AZIMUTHS    OF   THE    MOON.  371 

FIFTH    METHOD. — BY   AZIMUTHS    OF   THE    MOON,  OR    TRANSITS    OF    THE 
MOON   AND    A    STAR    OVER    THE    SAME    VERTICAL    CIRCLE. 

239.  The  travelling  observer,  pressed  for  time,  will  not  unfre^ 
quently  find  it  expedient  to  mount  his  transit  instrument  in  the 
vertical  circle  of  a  circumpolar  star,  without  waiting  for  the  meri- 
dian passage  of  such  a  star.     The  methods  of  determining  the 
local  time  and  the  instrumental  constants  in  this  case  are  given 
in  Vol.  II.     He  may  then  also  observe  the  transit  of  the  moon 
and  a  neighboring  star,  and  hence  deduce  the  right  ascension  of 
the   moon,  which   may  be   used  for  determining  his  longitude 
precisely  as  the  culminations  are  used  in  Art.  234. 

240.  But  if  the  local  time  is  previously  determined,  we  may 
dispense  with  all  observations  except  those  of  the  moon  and  the 
neighboring  star,  and  then  we  can  repeat  the  observation  several 
times  on  the  same  night  by  setting  the  instrument  successively 
in  different  azimuths  on  each  side  of  the  meridian.     It  will  not 
be  advisable  to  extend  the  observations  to  azimuths  of  more  than 
15°  on  either  side. 

The  altitude  and  azimuth  instrument  is  peculiarly  adapted  for 
such  observations,  as  its  horizontal  circle  enables  us  to  set  it  at 
any  assumed  azimuth  when  the  direction  of  the  meridian  is 
approximately  known.  The  zenith  telescope  will  also  answer 
the  same  purpose.  But  as  the  horizontal  circle  reading  is  not 
required  further  than  for  setting  the  instrument,  it  is  not  indis- 
pensable, and  therefore  the  ordinary  portable  transit  instrument 
may  be  employed,  though  it  will  not  be  so  easy  to  identify  the 
comparison  star. 

The  comparison  star  should  be  one  of  the  well-determined 
moon-culminating  stars,  as  nearly  as  possible  in  the  same 
parallel  with  the  moon,  and  not  far  distant  in  right  ascension, 
either  preceding  or  following. 

The  chronometer  correction  and  rate  must  be  determined,  with 
all  possible  precision,  by  observations  either  before  or  after  the 
moon  observations,  or  both.  An  approximate  value  of  the  cor- 
rection should  be  known  before  commencing  the  observations, 
as  it  will  be  expedient  to  compute  the  hour  angles  and  zenith 
distances  of  the  two  objects  for  the  several  azimuths  at  which  it 
is  proposed  to  observe,  in  order  to  point  the  instrument  properly 
and  thus  avoid  observing  the  wrong  star. 


372  LONGITUDE. 

To  secure  the  greatest  degree  of  accuracy,  the  observations 
should  be  conducted  substantially  as  follows : — 

1st.  The  instrument  being  supposed  to  have  a  horizontal  circle, 
let  the  telescope  be  directed  to  some  terrestrial  object,  the 
azimuth  of  which  is  known  (or  to  a  circumpolar  star  in  the  meri- 
dian), and  read  the  circle.  The  reading  for  an  object  in  the 
meridian  will  then  be  known  ;  denote  it  by  a. 

2d.  The  first  assumed  azimuth  at  which  the  transits  are  to  be 
observed  being  A,  set  the  horizontal  circle  to  the  reading  A  +  «, 
and  the  vertical  circle  to  the  computed  zenith  distance  of  the 
moon  or  the  star  (whichever  precedes).  This  must  be  done  a 
few  minutes  before  the  computed  time  of  the  first  transit. 

3d.  Observe  the  inclination  of  the  horizontal  axis  with  the 
spirit  level. 

4th.  Observe  the  transit  of  the  first  object  over  the  several 
threads. 

5th.  If  there  is  time,  observe  the  inclination  of  the  horizontal 
axis. 

6th.  Set  the  vertical  circle  for  the  zenith  distance  of  the  second 
object,  and  observe  its  transit. 

7th.  Observe  the  inclination  of  the  horizontal  axis  with  the 
spirit  level. 

The  instrument  must  not  be  disturbed  in  azimuth  during  these 
operations,  which  constitute  one  complete  observation. 

Now  set  upon  a  new  azimuth,  sufficiently  greater  to  bring  the 
instrument  in  advance  of  the  preceding  object,  and  repeat  the 
observation.  It  will  often  be  possible  to  obtain  in  this  way  four 
or  six  observations,  two  or  three  on  each  side  of  the  meridian, 
but  the  value  of  the  result  will  not  be  much  increased  by  taking 
more  than  one  observation  on  each  side  of  the  meridian. 

The  collimation  constant  is  supposed  to  be  known;  but,  in 
order  to  eliminate  any  error  in  it,  as  well  as  inequality  of  pivots, 
one-half  the  observations  should  be  taken  in  each  position  of 
the  rotation  axis. 

The  azimuth  of  the  instrument  at  each  observation  is  only 
known  from  the  local  time,  and  hence  the  following  indirect 
method  of  computation  will  be  found  more  convenient  than  the 
usual  method  of  reducing  extra-meridian  transits;  but  the 
reader  will  find  it  easy  to  adapt  the  methods  given  in  Yol.  II.  for 
such  purpose  to  the  present  case. 


BY   AZIMUTHS    OF    THE    MOON.  373 

We  shall  make  use  of  the  following  notation  : 

T,  T'  —  the  mean  of  the  chronometer  times  of  transit  of 
the  moon's  limb  and  the  star,  respectively,  over 
the  several  threads,* 
AT,  AT'  =  the  corresponding  chronometer  corrections, 

bj  b'  =  the  inclinations  of  the  horizontal  axis  at  the  times 

T  and  2", 
c  =  the   collimation    constant   for   the  mean   of  the 

threads. 

a,  a!  ==.  the  moon's  and  the  star's  right  ascensions, 
3,  3'  =  "  "  "      declinations, 

t,  t'   =  "  "  "       hour  angles, 

C,  C'  =  "  "  "      true  zenith  distances, 

q,  cf  =  "  "  "       parallactic  angles, 

A,A'  =  "  "  "       azimuths, 

Aa  —  the  increase  of  the  moon's  right  ascension  in  one 

minute  of  mean  time, 
&d  =  the  increase  (positive  towards  the  north)  of  the 

moon's  declination  in  one  minute  of  mean  time, 
TT  =  the  moon's  equatorial  horizontal  parallax, 
S  =  the  moon's  geocentric  sernidiameter, 
<p  =  the  observer's  latitude, 
Lr=  the  assumed  longitude, 
&L=  the  required  correction  of  this  longitude, 
L  =  the  true  longitude  =  U  -f  &L. 

The  moon's  a,  3,  TT,  and  S  are  to  be  taken  from  the  Ephemeris 
for  the  Greenwich  time  T  -{-  A  T  -\-  .//(expressed  in  mean  time). 
The  changes  AGC,  A d  are  also  to  be  reduced  to  this  time.  The 
right  ascension  and  declination  must  be  accurately  interpolated, 
from  the  hourly  Ephemeris,  with  second  differences. 

The  quantities  J.,  £,  q  are  now  to  be  computed  for  the  chro- 
nometer time  T,  and  A',  £',  qf  for  the  time  T' .  Since  A  and  Af 

*  The  chronometer  time  of  passage  over  the  mean  of  the  threads  will  be  obtained 
rigorously  by  reducing  each  thread  separately  to  the  mean  of  all  by  the  general 
formula  given  for  the  purpose  in  Vol.  II.  If,  however,  the  same  threads  are 
employed  for  both  moon  and  star,  and  c  denotes  the  equatorial  distance  of  the  mean 
of  the  actually  observed  threads  from  the  collimation  axis,  it  will  suffice  (unless  the 
observations  are  extended  greatly  beyond  the  limits  recommended  in  the  text)  to 
take  the  means  of  the  observed  times  at  the  times  of  passage  over  the  fictitious 
thread  the  collimation  of  which  is  —  c.  The  slight  theoretical  error  which  this 
procedure  involves  will  be  eliminated  if  the  observations  are  arranged  symmetrically 
with  respect  to  the  meridian. 


874 


LONGITUDE. 


are  required  with  all  possible  precision,  logarithms  of  at  least  six 
decimal  places  are  to  be  employed  in  their  computation ;  but  for 
C?  &  C'>  Q '>  f°ur  decimal  places  will  suffice.  The  following  formulae 
for  this  purpose  result  from  a  combination  of  (16)  and  (20) : 


For  the  moon. 
t=  T-f  AT—  a 

tan  M  =  tan  d  sec  t 

tan  t  cos  M 

tan  A  =  — 

sin  (<p  —  M.*}  . 


with  six 


decimals:    )  tan  A'  =  - 

'  CM 


For  the  star. 

if  =  T'-fAT'  —  a! 
{'=  tan  3'  sec  t' 
tan  t'  cos  M' 


sin  (<?  —  M') 


(426) 


tan  N  =  cot  tp  cos  t 
tan  t  ein  N 

inn    fi    

with  four 
decimals; 

Ltlll    U    

cos  (S  +  N) 
tm  r  -  cot  ^  +  *> 

ttlll    w    

cos  q 

tan  N'  =  cot  ^  cos  if 

tan  #'  sin  ^ 
tan  q  = 


tan  C'  = 


cos  (5'  -f-  JV') 
cot(3'  +  N') 


cos 


in  which  A  and  5-  are  to  be  so  taken  that  sin  A  and  sin  q  shall 
have  the  same  sign  as  sin  t. 

The  true  azimuth  of  the  moon's  limb  will  be  found  by  applying 
to  the  azimuth  of  the  centre  the  correction 

S    ["upper  sign  for  1st  limb"! 
~  sin  C  [lower     "      "    2d      "    J 

If  we  assume  the  parallax  of  the  limb  to  be  the  same  as  that  of 
the  centre  (which  involves  but  an  insensible  error  in  this  case), 
we  next  find  the  apparent  azimuth  of  the  limb  by  applying  the 
correction  given  by  (116),  or 

im(<p  —  </>')  sin  I"  sin  A'  cosec  C 

in  which  <p  —  cp'  is  the  reduction  of  the  latitude,  and  />  is  the 
terrestrial  radius  for  the  latitude  <p.  In  this  expression  we 
employ  A,  which  is  the  computed  azimuth  of  the  star,  for  the 
apparent  azimuth  of  the  moon's  limb,  since  by  the  nature  of  the 
observation  they  are  very  nearly  equal. 

To  correct  strictly  for  the  collimation  and  level  of  the  instru- 
ment, we  must  have  the  moon's  and  star's  apparent  zenith  dis- 
tances, which  will  be  found  with  more  than  sufficient  accuracy 
for  the  purpose  by  the  formulae 


moon's  app.  zen.  dist.  =  Ci  =  C  ~f  *  sin  C  —  refraction 
star's        "       "       "     =Ci'=  C'—  refraction 


BY   AZIMUTHS    OF    THE    MOON.  375 

and  then  the  reduction  of  the  true  azimuth  to  the  instrumental 
azimuth  (see  Vol.  II.,  Altitude  and  Azimuth  Instrument)  is 

j, 
for  the  moon, 


for  the  star,       =p  - 


sin  Ci       tan  d 
c  V 


tan 


the  upper  or  lower  sign  being  used  according  as  the  vertical 
circle  is  on  the  left  or  the  right  of  the  observer.  The  computed 
instrumental  azimuths  are,  therefore, 


S        px(<f> — <p')  sin  1"  sin  A'         c  b 

(moon)  Ax  =A  ±- 


"sin  C  sin  C  sinCi       tan  Ci 

• 

c  V 

(star)   Ai=A'+—      -=F- 

smC/      tanC/ 


(427) 


If  now  the  longitude  and  other  elements  of  the  computation  are 
correct,  we  shall  find  Al  and  J./  to  be  equal:  otherwise,  put 

x  =  A,  —  A{  (428) 

then  we  are  to  find  how  the  required  correction  AZ/  depends  on  z, 
supposing  here  that  all  the  elements  which  do  not  involve  the 
longitude  are  correct.  Now,  we  have  taken  a  and  d  from  the 
Ephemeris  for  the  Greenwich  sidereal  time  T  -j-  A  T  -f-  L',  when 
they  should  be  taken  for  the  time  T+  &T  -{-  Lf+  &L.  Hence, 
if  /I  and  /9  denote  the  increments  of  the  moon's  right  ascension 
and  declination  in  one  sidereal  second,  both  expressed  in  seconds 
of  arc, 

15  Aa    _ 

60.164  (429) 

^6ok-[8-22066]^ 

we  find  that 

a  requires  the  correction     A .  &L 
d        "  «  /S.Ai 

t        «  "         -A.A£ 

and  these  corrections  must  produce  the  correction  —  x  in  the  moon's 
azimuth.  The  relations  between  the  corrections  of  the  azimuth, 
the  hour  angle,  and  the  declination,  where  these  are  so  small  as 
to  be  treated  as  differentials,  is,  by  (51), 


376  LONGITUDE. 

dA=       c 

sin  C  sin  C 

that  is, 

cos  d  cos  <?        ...      sin  q 
—  x  =  --          —  i  I  .&L  -f       - 
sin  C  •  sin  C 

Hence,  if  we  put 


sin  C  sin  C 

we  have 


A£  =  -  (431) 

and  hence,  finally,  the  true  longitude  Lfjr  &L. 

241.  In  order  to  determine  the  relative  advantages  of  this 
method  and  that  of  meridian  transits,  let  us  investigate  a  formula 
which  shall  exhibit  the  effect  of  every  source  of  error.  Let 

<?a,  3d,  dx,  dS  =  the  corrections  of  the  elements  taken  from 

the  Ephemeris  of  the  moon, 
da,  ddf  =  the  corrections  of  the  star's  place, 
dTj  5Tr  =  the  corrections  for  error  in  the  obs'd  time, 
£A  T  =  the  correction  of  A  T, 
d<p  =  the  correction  of  <p. 

If,  when  the  corrected  values  of  all  the  elements  —  that  of  the 
longitude  included—  are  substituted  in  the  above  computation, 
Al  and  AJ  become  Al  +  dA^  and  AJ  -\-  dA^  we  ought  to  find, 
rigorously, 

A,  -f  dAi  =  AS  -f  dAJ 

which  compared  with  (428)  gives 

x  =  —  dAl  -f-  dA,f  (432) 

We  have,  therefore,  to  find  expressions  for  dA^  and  dA±  in 
terms  of  the  above  corrections  and  of  &L.  We  have,  first,  by 
differentiating  (427), 


dAl  =  dA  + 


sinC  sin  ^ 


We  neglect  errors  in  c  and  b  which  are  practically  eliminated 
by  comparing  the  moon  with  a  star  of  nearly  the  same  declina- 
tion, and  combining  observations  in  the  reverse  positions  of  the 
axis. 


BY    AZIMUTHS    OF    THE    MOON.  377 

The  total  differential  of  A  is,  by  (51),  after  reducing  dt  to  arc, 

cos  d  cos  a    .  ,    sin  q  _ 

dA  =  -          —  ^-  •  15  <ft  -|-       -  dd  —  cot  C  sin  A  d<p 
sin  C  sin  C 

consequently,  also, 

...        cos  d'  cos  q'  ,         sin  q'  . 

dA'  —  —         —  —  •  l&dr  -\  —    —dd'  —  cot  C   sm  A'dw 
sinC'  sinC' 

Since  t  =  T  +  A  7"  —  a,  we  have 

dt  =  dT       d±T  —  da, 


where  dT  and  C?A  J'may  be  at  once  exchanged  for  £  J'and 
but  doc  is  composed  of  two  parts  :  1st,  the  correction  of  the 
Ephemeris,  and  2d,  X(*L  +  dT+  d*T),  which  results  from  our 
having  taken  a  for  the  uncorrected  time.  Hence  we  have,  in 
arc, 

15  3&T—  lo  da  —  AA£  -     dT 


The  correction  dd  is  likewise  composed  of  two  parts,  namely, 

dd  =  M  -f-/5(Ai-f  dT+  ^AT) 
Further,  we  have  simply  dd'  —  odf  and 

—  da' 


but,  as  we  may  neglect  the  error  in  the  rate  of  the  chronometer 
for  the  brief  interval  between  the  observation  of  the  moon  and 
the  star,  we  can  take  OA  T'  =  3&  T,  and,  consequently, 

When  the  substitutions  here  indicated  are  made  in  (432),  we 
obtain  the  expression 


sin  C 

^.W+15/'. 

sin  C 

"B[nA'dx 


sin  C  sin  C 

[15 (/ - /')  -  a\  <JA T  +  — (^ ~C)sin^  ^  (433) 

sin  C  sin  C' 


378  LONGITUDE. 

in  which  the  following  abbreviations  are  used  : 

cos  3  cos  q  cos<5'cos<?' 


smC 

and  in  the  coefficient  of  d<p  we  have  put  A  =  A'. 

By  the  aid  of  this  equation  we  can  now  trace  the  effect  of 
each  source  of  error. 

1st.  The  coefficients  of  00,  odf,  O/T,  o<p  have  different  signs  for 
observations  on  different  sides  of  the  meridian,  and  therefore 
the  errors  of  decimation,  parallax,  and  latitude  will  be  elimi- 
nated by  taking  the  mean  of  a  pair  of  observations  equidistant 
from  the  meridian. 

2d.  The  star's  declination  being  nearly  equal  to  that  of  the 
moon,  we  shall  have  very  nearly/  —  /',  and  the  coefficient  of 
^A  Twill  be  —  a;  and  since  to  lintl  &L  we  have  yet  to  divide 
the  equation  by  «,  it  follows  that  an  error  in  the  assumed  clock 
correction  produces  an  equal  error  (but  with  a  different  sign)  in 
the  longitude,  as  in  the  case  of  meridian  observations. 

3d.  An  error  dT  in  the  observed  time  of  the  moon's  transit 
produces  in  the  longitude  the  error 


The  mean  of  the  values  of  a  for  two  observations  equidistant 
from  the  meridian  is  Xf.     The  mean  effect  of  the  error  d  T  is 

therefore 

/  15         \ 
-—  1\3T 


which  is  the  same  as  in  the  case  of  a  meridian  observation. 

The  effect  of  an  error  d  Tf  in  the  observed  time  of  the  star's 
transit  is 


and  for  two  observations  equidistant  from  the  meridian,  the  star 
being  in  the  same  parallel  as  the  moon,  the  mean  effect  is 

TST< 

also  the  same  as  for  a  meridian  observation. 


BY   AZIMUTHS    OF   THE    MOON.  379 

4th.  An  error  SS  in  the  tahular  semidiameter  is  always  elimi- 
nated in  the  case  of  meridian  observations  when  they  are  com- 
pared with  observations  at  another  meridian,  since  the  same 
semidiameter  is  employed  in  reducing  the  "observations  at  both 
meridians.  But  in  the  case  of  an  extra-meridian  observation  the 
effect  upon  the  longitude  is 

dS  3S 


and   in   the   mean   of  two   observations   equidistant  from  the 
meridian,  the  values  of  q  being  small,  it  is 

dS    (1  +  2  sin1  J  0)  nearly. 


>l  cos<S  cosq       >icos«5 
For  a  meridian  observation  the  error  will  be 


A  cos  <5 

The  error  in  the  case  of  extra-meridian  observations,  therefore, 
remains  somewhat  greater  than  in  the  case  of  meridian  ones,  the 
excess  being  nearly 

2dS.sin.nq 
A  cos  d 

which,  however,  is  practically  insignificant ;  for  we  have  not  to 
fear  that  dS  can  be  as  great  as  1",  and  therefore,  taking  q  =  15°, 
d  =  30°,  and  /I  =  0.4,  which  are  extreme  values,  the  difference 
cannot  amount  to  O'.l  in  the  longitude. 

5th.  The  error  da  of  the  tabular  right  ascension  of  the  moon 
produces  in  the  longitude  the  error 

16/. 

-  da 

a 

and  from  the  mean  of  two  observations  equidistant  from  the 
meridian,  the  error  is 

15  tig 
A 

as  in  the  case  of  the  meridian  observation. 

The  error  da'  in  the  star's  right  ascension  produces  the  error 

— —  when  the  star  is  in  the  same  parallel  as  the  moon. 


380  LONGITUDE. 

From  this  discussion  it  follows  that,  by  arranging  the  observa- 
tions symmetrically  with  respect  to  the  meridian,  the  mean  result 
will  be  liable  to  no  sensible  errors  which  do  not  equally  affect 
meridian  observations.  But  for  the  large  culmination  error  in 
the  case  of  the  moon  (Art.  236),  which  equally  affects  extra- 
meridian  observations,  the  latter  would  have  a  great  advantage 
by  diminishing  the  effect  of  accidental  errors.  But  the  probable 
error  of  the  mean  of  two  observations  equidistant  from  the 
meridian,  seven  threads  being  employed,  will  be,  by  (422), 


and  that  of  a  single  meridian  observation,  even  where  only  one  star 
is  compared  with  the  moon,  is,  by  the  same  formula,  =  OM1.  When 
we  take  into  account  the  extreme  simplicity  of  the  computation, 
the  method  of  moon  culminations  must  evidently  be  preferred  ; 
and  that  of  extra-meridian  observations  will  be  resorted  to  only 
in  the  case  already  referred  to  (Art.  239),  where  the  traveller 
may  wish  to  determine  his  position  in  the  shortest  possible  time 
and  without  waiting  to  adjust  his  instrument  accurately  in  the 
meridian. 

EXAMPLE.  —  At  the  U.  S.  Naval  Academy,  1857  May  9,  I  ob- 
served the  following  transits  of  the  moon's  second  limb  and  of 
a  Scorpii,  at  an  approximate  azimuth  of  10°  East,  with  an  ERTEL 
universal  instrument  of  15  inches  focal  length  : 

Chronometer.  Level.          Collim. 

3  II  Limb.  T  =  16*  11™  3(K17  b  =  -f  2".2  c  =  0.0)  Vertical  circle 
c  Scorpii  T'=16  27  49.83  V=  +  2  .2  3  *left. 

These  times  are  the  means  of  three  threads.     The  chronometer 
correction,   found   by  transits   of    stars   in   the   meridian,   was 
-  55m  9M6  at  13A  sidereal  time,  and  its  hourly  rate  —  09.32.    The 
assumed  latitude  and  longitude  were 

V  =  38°  58'  53".5  L'  =  5*  5-  55' 

The  star's  place  was 

a'  =  16*  12-  31-.90  3'=  —  25°  14'  58".5 


BY   AZIMUTHS    OF    THE    MOOX. 


381 


We  first  find  the  sidereal  times  of  the  observations  of  the 
moon  and  star  respectively,  and  the  Greenwich  mean  time  of 
the  observation  of  the  moon  :  we  have 


=  —   55-    9'.97 


—  55-    9-.S9 
T-f  AT  =  15*16-20'.28 

L'  =    5  5    55. 

Gr.  sidereal  time  =  20  22    15  .28 

Sid.  time  Gr.  moon  =    3  8    58  .91 

Sidereal  interval  =  17  13    16  .37 

Red.  to  mean  time  =  —  2    49  .28 


Gr.  mean  time  ==  May  9, 17*  10-  278.09 
Hence  from  the  Ephemeris  we  find 
a  =  15*  54-  45'.32 

Aa  =  2M135 

S  =  14'  47".2 


3  =  —  24°  42'  54".4 

<5  ==  —  7".619 
7c=       54'    9".2 


By  (426)  we  find 

A  =  —  9°  40'  51".0 

log  sin  q  =  n9.1581 

C  ==  64°  19'.5 

TT  sin  C  =  +48  .8 

Refraction  =  2  .1 

C-.se  65      6 .2 


A'  =  —    9°  57'  14".8 
log  sin  q'  =       n9.1719 
C'  =       64°  54M 


Refraction  = 


—     2.1 
64    52.0 


For  the  latitude  y  we  find,  from  Table  III., 

log  P  =  9.9994  <p  —  ?'==  11'  15' 

and  then,  by  (427),  we  find 

A  =  —  9°  40'  51".0 

8 

-       16  24  .4 


sin  C 
—  ?>')  sin  1"  sin  J.' 


sn 


tan 


2  .0 

0  .0 

1  .0 


Ll=:_9    57  18  .4 


=  —  9°  57' 


sin  :/ 


tan  C/ 


0  .0 

1  .0 


{  =  —  9    57  15  .8 


382  LONGITUDE. 

whence 

x  =  —  2".6 

By  (429),  (430),  and  (431),  we  find 

log  A  —  9.72175  log  /?  =  w9.10256  a  =  0.5054 


0.5054 

If  we  wish  to  see  the  effect  of  all  the  sources  of  error  in  this 
example,  we  find,  by  (433), 

0.5054  A/v  =  —  2".  6  —  14.96  <Sa  —  0.1G  <M  +  14.45  AT  —  14.82  AT'  —  0.36  A&T 
4-14.82  rfa'  -f-0.16  t<Y  4-    1.11  AS  —  0.001  fa    -f-  0.002  <ty 

The  proper  combination  of  observations  is  supposed  to  eliminate, 
or  at  least  reduce  to  a  minimum,  all  the  errors  except  that  of  the 
moon's  right  ascension  as  given  in  the  Ephemeris.  In  practice, 
therefore,  it  will  be  necessary  to  retain  the  term  involving  da. 
Thus,  in  the  present  case  we  take  only 

0.5054  *•£=:  —  2".6  —  14.96  &» 

A   second   observation  on  the  same  day  at  an  azimuth   10° 

west  gave 

0.5458  A£  =  —  5".7  —  14.92  <5a 

The  elimination  of  the  errors  of  declination  requires  that  we 
take  the  arithmetical  mean  of  these  equations;  whence  we  have, 

finally, 

A  L  =  —  7'.89  —  28.43  <?a 

SIXTH    METHOD.  —  BY   ALTITUDES    OF    THE    MOON. 

242.  The  hour  angle  (t)  of'  the  moon  may  be  computed  from 
an  observed  altitude,  the  latitude  and  declination  being  known, 
and  hence  with  the  local  sidereal  time  of  the  observation  (—  0) 
the  moon's  right  ascension  by  the  equation  a  =  0  —  £,  with 
which  the  Greenwich  time  can  be  found,  as  in  Art.  234,  and, 
consequently,  also  the  longitude. 

The  hour  angle  is  most  accurately  found  from  an  altitude 
when  the  observed  body  is  on  the  prime  vertical,  and  more 
accurately  in  low  latitudes  than  in  high  ones  (Art.  149).  This 
method,  therefore,  is  especially  suited  to  low  latitudes. 

The  method  maybe  considered  under  two  forms:  —  (A)  that  in 
which  the  moon's  absolute  altitude  is  directly  observed  and 


BY    ALTITUDES    OF    THE    MOON.  383 

employed  in  the  computation  of  the  hour  angle  ;  and  (B)  that  in 
which  the  moon's  altitude  is  compared  differentially  with  that  of 
a  neighboring  star, — i.e.  when  the  moon  and  a  star  are  observed 
either  at  the  same  altitude,  or  at  altitudes  which  differ  only  by  a 
quantity  which  can  be  measured  with  a  micrometer. 

243.  (A.)  By  the  moon's  absolute  altitude. — This  method  being 
practised  only  with  portable  instruments,  it  would  be  quite 
superfluous  to  employ  the  rigorous  processes  of  correcting  for 
the  parallax,  which  require  the  azimuth  of  the  moon  to  be  given. 
The  process  of  Art.  97  will,  therefore,  be  employed  in  this  case 
with  advantage,  by  which  the  observed  zenith  distance  is  reduced 
not  to  the  centre  of  the  earth,  but  to  the  point  of  the  earth's 
axis  which  lies  in  the  vertical  line  of  the  observer,  arid  which 
we  briefly  designate  as  the  point  0.  Let 

C"  =  the  observed  zenith  distance,  or  complement  of  the 

observed  altitude,  of  the  moon's  limb, 
0  =  the  local  sidereal  time, 
Lf  =  the  assumed  longitude, 
A.Z/  =  the  required  correction  of  L', 
L  =  the  true  longitude  ='L'  -j-  &L. 

Find  the  Greenwich  sidereal  time  ©  -f  Lr,  and  convert  it  into 
mean  time,  for  which  take  from  the  Ephemeris  the  quantities 

d  =  the  moon's  declination, 

•K  =          "  eq.  hor.  parallax, 

S  =          "  semidiameter. 

Let  Sf  be  the  apparent  semidiameter  obtained  by  adding  to  S 
the  augmentation  computed  by  (251)  or  taken  from  Table  XII. 
Let  r  be  the  refraction  for  the  apparent  zenith  distance  £" ;  and 
put 

C'=C"-f  r±S'  (434) 

Let  TTj  be  the  corrected  parallax  for  the  point  0,  found  by  (127), 
or  by  adding  to  n  the  correction  of  Table  XIH.  (which  in  the 
present  application  will  never  be  in  error  0".l) ;  and  put 

d.  =  d  4-  e2  TT.  sin  <p  cos  5 
C;  =  £'-*,  sin/ 

in  which  log  e2=  7.8244. 


384  LONGITUDE. 

The  hour  angle  (which  is  the  same  for  the  point  0  as  for  the 
centre  of  the  earth)  is  then  found  by  (267),  i.e. 


=     l( 

\   \ 


(436) 


COS  <f>  COS  dl 

after  which  the  moon's  right  ascension  is  found  by  the  formula 

a  =  0  —  t  (437) 

and  hence  the  Greenwich  time  and  the  longitude  as  above  stated. 
But  since  we  have  taken  d  for  an  approximate  Greenwich  time 
depending  on  the  assumed  longitude,  the  first  computation  of  t 
will  not  be  quite  correct  ;  a  second  one  with  a  corrected  value 
of  d  will  give  a  nearer  approximation;  and  thus  by  successive 
approximations  the  true  value  of  t  and  of  the  longitude  will  at 
last  be  found. 

But  instead  of  these  successive  approximations  we  may  obtain 
at  once  the  correction  of  the  assumed  longitude,  as  follows.  We 
have  taken  d  for  the  Greenwich  time  0  +  L  ',  when  we  should 
have  taken  it  for  the  time  0  -f  Lr  +  &L.  Hence,  putting 

/?  =  the  increase  of  d  in  a  unit  of  time, 

it  follows  that  d  requires  the  correction  ft&L;  and  therefore,  by 
(51),  the  correction  of  the  computed  hour  angle  will  be 


cos<5 


in  which  q  is  the  parallactic  angle.     Since  a  =  0  —  t,  the  com 
puted  right  ascension  requires  the  correction  (in  seconds  of  time) 


15  cos  d  t&nq 
Therefore,  if  we  put 

X  =  the  increase  of  a  in  a  unit  of  time, 

the  computed  Greenwich  time  and,  consequently,  also  the  longi 
tude  derived  from  it  requires  the  correction 


15  A  cos  d  tan  q 


BY   ALTITUDES    OF    THE    MOON. 


385 


Hence,  denoting  the  longitude  computed  from  the  right  ascen- 
sion a  =  @  —  t  by  L",  we  have 


True  longitude  =  L'  -|-  ±L  =  L"  — 


15  A  cos  <S  tan  q 


whence 


L"—L 


1  + 


If  we  denote  the  denominator  of  this  expression  by  1  +  a,  we 
shall  have,  by  (18), 

(2    /  tan  <p        tan  3  \ 

d  ^^  •    _    .'  I  — : I 


and  then 


15  A  \  sin  t 
L"—L' 


(438) 


(439) 


EXAMPLE. — At  the  U.  S.  Naval  Academy,  in  latitude  <p  =  38° 
58'  53"  and  assumed  longitude  L'=  5h  6m  0*,  I  observed  the 
double  altitude  of  the  moon's  upper  limb  with  a  sextant  and 
artificial  horizon  as  below : 

1849  May  2. — Moon  east  of  the  meridian. 


Chronometer 

10»  14«  2K6 

Mean  of  6  obs.  2 

& 

=     64° 

40' 

0" 

Fast 

4 

41 

0.0 

Index  corr.  of  sextant  =     — 

14 

57 

Local  mean  time  = 

5 

33 

21.6 

2)64 

25 

3 

Assumed  L'          — 

5 

8 

0. 

alt  ~j) 

—     32 

1° 

31    5 

10 

39 

21.6 

Approx.  Gr.  time  — 

=     57 

47 

28  .5 

(For  which  time  we 
£  from  the  Nautical 

take 
Alma 

IT,  S,  and 
nac) 

Barom.                     30«».45 
Att.  Therm.             63°  F. 
Ext.     «                   65°  F. 

} 

r 

1 

30  .9 

t  J  wl  sin  <f>  cos  8  = 

+  3°  47' 

47".6 
14  .1 

AS  (Tab. 

XII.)  =     +8 

•11 

S' 

=     + 

15 

24  .5 

+  3 

48 

1  .7 

V 

=     58 

4 

23  .9 

AJT  (Tab. 

xm.)  =  +   4 

iu 

sin<r' 

= 

47 

38  .1 

TT,  =  56    7 

.5) 

f, 

=     57 

16 

45  .8 

With  these  values  of  dv  f  p  and  y  ==  38°  58r  53",  we  find,  by  (436), 
t  =  —  3*  19"  53'.64 

The  sidereal  time  at  Greenwich  mean  noon,  1849  May  2,  was 
2*4i»  7*.98;  whence 

0=8*  16-  14-.61 
a  =  11   36      8  .25 

VOL.    L— 25 


386  LONGITUDE. 

Corresponding  to  this  right  ascension  we  find  by  the  hourly 
Ephemeris  the  Greenwich  mean  time,  and  hence  the  longitude 
-L",  as  follows  : 

Greenwich  mean  time  =  10*  39m  48*.  7 

Local  «        "     —    5   33    21  .6 

£"  =  ~5     6    27.1 

L"—L'  =        +    27M 

By  the  hourly  Ephemeris  we  also  have  for  the  Greenwich  time 

10*  39-  48*.7, 

Increase  of  a  in  lm  =  A  =  -f-  2'.014 


and  hence,  by  (438)  and  (439), 

a  =  —  0.3317  Ai  =  +  40-.6 

L  =  L'  -f  A£  ==  5*  6*  40-.6 

244.  The   result   thus   obtained   involves  the   errors   of   the 
tabular  right  ascension  and  declination    and   the   instrumental 
error.    The  tabular  errors  are  removed  by  means  of  observations 
of  the  same  data  made  at  some  of  the  principal  observatories,  as 
in  the  case  of  moon  culminations.     The  instrumental  error  will 
be  nearly  eliminated  by  determining  the  local  time  from  a  star 
at  the  same  altitude  and  as  nearly  as  possible  the  same  declina- 
tion ;    for   the   instrumental   error  will  then  produce  the  same 
error  in  both  0  and  t,  and,  therefore,  will  be  eliminated  from 
their  difference  0  —  t  =  a.     The  error   in   the   longitude  will 
then  be  no  greater  than  the  error  in  0.     But  to  give  complete 
•effect  to  this  mode  of  eliminating  the  error,  an  instrument,  such 
as  the  zenith  telescope,  should  be  employed,  which  is  capable  of 
indicating  the  same  altitude  with  great  certainty  and  does  not 
involve   the    errors  of  graduation  of  divided  circles.     A  very 
different  method  of  observation  and  computation  must  then  be 
resorted  to,  which  I  proceed  to  consider. 

245.  (B.)  ]3y  equal  altitudes  of  the  moon  and  a  star,  observed  with 
the  zenith  telescope.  —  The  reticule  of  this  instrument  should  for 
these  observations  be  provided  with  a  system  of  fixed  horizontal 
threads  :  nevertheless,  we  may  dispense  with  them,  and  employ 
only  the  single  movable  micrometer  thread,  by  setting  it  suc- 
cessively at  convenient  intervals. 


BY   ALTITUDES    OF   THE    MOON.  387 

Having  selected  a  well  determined  star  as  nearly  as  possible 
in  the  moon's  path  and  differing  but  little  in  right  ascension,  a 
preliminary  computation  of  the  approximate  time  when  each 
body  will  arrive  at  some  assumed  altitude  (not  less  than  10°) 
must  be  made,  as  well  as  of  their  approximate  azimuths,  in 
order  to  point  the  instrument  properly.  The  instrument  being 
pointed  for  the  first  object,  the  level  is  clamped  so  that  the 
bubble  plays  near  the  middle  of  the  tube,  and  is  then  not  to  be 
moved  between  the  observation  of  the  moon  and  the  star.  After 
the  object  enters  the  field,  and  before  it  reaches  the  first  thread, 
it  may  be  necessary  to  move  the  instrument  in  azimuth  in  order 
that  the  transits  over  the  horizontal  threads  may  all  be  observed 
without  moving  the  instrument  during  these  transits.  The  times 
by  chronometer  of  the  several  transits  are  then  noted,  and  the 
level  is  read  off.  The  instrument  is  then  set  upon  the  azimuth 
of  the  second  object,  the  observation  of  which  is  made  in  the 
same  manner,  and  then  the  level  is  again  read  off.  This  com- 
pletes one  observation.  The  instrument  may  then  be  set  for 
another  assumed  altitude,  and  a  second  observation  may  be  taken 
in  the  same  manner.*  Each  observation  is  then  to  be  separately 
reduced  as  follows  :  Let 

i,  ir,  i",  &c.  =  the  distances   in  arc  of  the   several   threads 

from  their  mean, 

m,  m'  =  the  mean  of  the  values  of  i  for  the  observed 
threads,  in   the   case  of  the    moon    and    star 
respectively, 
1,1' =  the  level  readings,  in  arc,  for  the  moon  and 

star, 

0,  0'=  the  mean  of  the  sidereal  times  of  the  observed 
transits  of  the  moon  and  star; 

then  the  excess  of  the  observed  zenith  distance  of  the  moon's 
limb  at  the  time  0  above  that  of  the  star  at  the  time  0'  isf 

m  —  m'  +  I  —  I' 

the  quantities  m  and  I  being  supposed  to  increase  with  increasing 
zenith  distance. 

*  The  same  method  of  observation  may  be  followed  with  the  ordinary  universal 
instrument,  but,  as  the  level  is  generally  much  smaller  than  that  of  the  zenith  tele- 
scope, the  same  degree  of  accuracy  will  not  be  possible. 

f  When  the  micrometer  is  set  successively  upon  assumed  readings,  m  and  m'  will 
be  the  means  of  these  readings,  converted  into  arc,  with  the  known  value  of  the  screw. 


388 


LONGITUDE. 


AlfiO,  let 

a,  d,  t,  C,  A,  q  =  the  E.  A.,  decl.,  hour   angle,  geocentric 
zenith  distance,  azimuth,  and  pnrallactic 
angle  of  the  moon's  centre  at  the  time 
0; 
a',  d',  f,  C',  A',  q'=  the  same  for  the  star  at  the  time  0'; 

TT,  8  —  the  moon's  equatorial  hor.  parallax  and 

semidiameter; 
A  —  the  increase  of  a  in  1*  of  sid.  time  ; 


«         u 


^  =  the  latitude; 
L'  =  the  assumed  longitude; 
&L  =  the  required  correction  of  L'  ; 

The  quantities  a,  <5,  TT,  and  $  are  to  be  taken  from  the  Ephemeris 
for  the  Greenwich  sidereal  time  0  -f-  Lf  (converted  into  mean 
time)  ;  a  and  3  being  interpolated  with  second  differences  by  the 
hourly  Ephemeris.  Then  the  required  correction  of  the  longi- 
tude will  be  found  by  comparing  the  computed  value  of  £  with 
the  observed  value.  For  this  purpose  we  first  compute  £  and  £' 
with  the  greatest  precision,  and  also  A  and  q  approximately.  If 
the  differential  formula  of  the  next  article  is  also  to  be  computed, 
Af  and  qr  will  also  be  required.  The  most  convenient  formulae 
will  be  — 


For  the  moon. 
t  =  0  —  a 

tan  M=t&n  d  secf 


cos  C=. 


—  M) 


For  the  star. 
f=  0'—  a' 

rtanJf'=tan  8'  sec  if 
with  six 


sin  M  • 


,     .  <          „, 

decimals;   )    cos  C  — 


—  M'  ) 


cos  A  =  tan  (y>  —  M  )  cot  C 
taniV=cot  <f>  cos  t 

tan  t  sin  JV 


cos  (d  -f  N) 


with  four 
decimals; 


amM' 

(440) 

cos  A'  =  tan  (<p  —  M'")  cot  C ' 
tan  N'  =  cot  <p  cos  if 

tdtf/sfajV 

tan  q  =  — 

*•  _  ^ «/    i      ^r/\ 


The  zenith  distance  £  thus  computed  will  not  strictly  correspond 
to  the  time  0  unless  the  assumed  longitude  is  correct.  Let  its 
true  value  be  £  +  d£.  Also  put 

Cj  =  the  observed  zenith  distance  of  the  moon's  limb, 
C,'  =  the  observed  zenith  distance  of  the  star, 
r,  rf  =  the  refraction  for  r,  and  r'. 


BY   ALTITUDES    OF   THE    MOON. 


389 


then 


t  =  C/  +  m  —  m'  +  I  —  lr 


Putting  then 

C"=  Ci  +  r  =  C'  +  m  -  m'  +  Z  -  T  +  (r  -  r') 
and,  by  Art.  (136), 

f  ==.  (<p  —  <p')  cos  A  sin/>— /?sin7T8in(C" — 

k  =  p  =p  $  q=  J  (_p  =p  $)  sin  ^)  sin  $ 


(441) 


the  <  ,  *         rsign  being  used  for  the  moon's  s  ,  >  limb, 

( lower  j  t  lower  j 


have 


This  equation  determines  rff.  We  have,  therefore,  only  to 
determine  the  relation  between  d£  and  AZ/.  Now,  we  have  taken 
a  and  d  for  the  Greenwich  sidereal  time  0  -f  L ',  when  we  should 
have  taken  them  for  the  time  0  -f  Lf  -j-  Ai':  hence 


a  requires  the  correction 


and  then,  by  (51), 

d?  =  —  cos  q  .  p&L  —  sin  q  cos  d .  15 

Hence,  putting  x  =  —  d£,  or 

~, >• *•//   i    j, 

a  =  15  A  sin  g  cos  5  -f-  /?  cos  q 


and 
we  have 


^ 

Ai  —  — 

a 


(442) 


The  solution  of  the  problem,  upon  the  supposition  that  all  the 
data  are  correct,  is  completely  expressed  by  the  equations  (440), 
(441),  and  (442). 

246.  The  quantity  x  is  in  fact  produced  not  only  by  the  error 
in  the  assumed  longitude,  but  also  by  the  errors  of  observation 
and  of  the  Ephemeris.  In  order  to  obtain  a  general  expression 


390  LONGITUDE. 

in  which  the  effect  of  every  source  of  error  may  be  represented, 
let 

T,  Tf  =  the  chronometer  times  of  observation  of  the 

moon  and  star, 

A  T  =  the  assumed  chronometer  correction, 
dT,  dT'=  the  corrections  of   T  and  T'  for  errors  of 

observation, 

dA  T  =  the  correction  of  A  T, 
o'a,  dd,  dn,  dS  =  the  corrections  of  the  elements  taken  from 

the  Ephemeris, 
d(f>  =  the  correction  of  the  assumed  latitude. 

If,  when  the  corrected  values  of  all  the  elements  are  substituted, 
£,  £',  k  become  £  +  </£,  £  -f-  </£',  k  +  c?Ar,  instead  of  the  equation 
C"  —  (C  +  <^C)  =  ^  we  shall  have 

C''+dC'—(C  +  #)  =  *  +  <» 

and  hence 

x  =  —  dZ  +  d:'  —  dk  (443) 

and  we  have  now  to  find  expressions  for  </£,  d£'9  and  c?A;  in  terms 
of  the  above  corrections  of  the  elements. 

Taking  all  the  quantities  as  variables,  we  have 

d*  =  15  sin  q  cos  3  dt  —  cos  q  ctt  -f  cos  A  d<p 
dZ'=  15sin</'cos  8'  dt'  —  cos  q'  do'  -{-  cos  A'  d<p 

Since  t  =  T  -f  A  T  —  a,  we  have 

dt  =  dT  +  d&T  —  do, 

where  dT  &ud  d&Tm&j  be  exchanged  for  dTsmd  S&T,  but  da  is 
composed  of  two  parts  :  1st,  of  the  actual  correction  of  the 
Ephemeris;  and  2d,  of  /(A£  +  dT-}-  S&T)  resulting  from  our 
having  taken  a  for  the  unconnected  time  :  hence  we  iiave 


dt  =  ST  -\-  d*  T—  oa  —  A(AZ  -f  dT+  d&T) 
The  correction  dd  is  also  composed  of  two  parts,  so  that 

dd  =  dd  4-  /5(Ai  -f-  oT  -}-  d  AT) 

Further,  we  have  simply  do'  =  So',  and 

dtf=  ST'+  d^T—  oV 
in  which  ^A  T  at  the  time  T'  is  assumed  to  be  the  same  as  at  the 


BY   ALTITUDES    OF    THE    MOOX.  391 

time  T7  an  error  in  the  rate  of  chronometer  being  insensible  in 
the  brief  interval  between  the  observations  of  the  moon  and  the 
star. 
Again,  we  have,  from  (441), 

cos  p  dp  =  p  cos  TT  sin  (C"  —  y)  d*  -f-  p  sin  TT  cos  (C"  —  y)  d'" 
dk  =  dp  =p  d$ 

or,  with  sufficient  accuracy, 

dk  =  sin  C'  <5*  qp  <S$  -f-  sin  ?r  cos  C'  d£' 

Now,  substituting  in  rf£  and  rf£'  the  values  of  dt,  dd,  &e.,  and  then 
substituting  the  values  of  d£  and  <:/£'  thus  found,  in  (443),  together 
with  the  value  of  dk,  we  obtain  the  final  equation  desired,  which 
may  be  written  as  follows  :* 


x  =  a&L  -\-f .  da  -f-  cos  q  .  3d  —  (/  —  a)  dT 

—  mf .  da,'  —  m  cos  q'  dd'  -f-  mf  .  8T' 
±  dS —  siri  C'  fa  —  (/ —  mf  — 

—  (cos  A  —  m  cos  A')  dp 

where  the  following  abbreviations  are  employed: 

f  =  15  sin  q  cos  3  f  =  15  sin  q'  cos  3' 

a  =  If  -\-  0  cos  q  m  =  1  —  sin  TT  cos  C' 

Having  computed  the  equation  in  this  form,  every  term  is  to 
be  divided  by  a,  and  then  &L  will  be  obtained  in  terms  of  x  and 
all  the  corrections  of  the  elements. 

A  discussion  of  this  equation,  quite  similar  to  that  of  (433), 
will  readily  show  that  the  observations  will  give  the  best  result 
when  taken  near  the  prime  vertical  and  in  low  latitudes,  and. 
farther,  that  the  combination  of  observations  equidistant  from 
the  meridian,  east  and  west,  eliminates  almost  wholly  errors  of 
declination  and  parallax  and  of  the  chronometer  correction. 

EXAMPLE.-)-— At  Batavia,  on  the  llth  of  October,  1853,  Mr. 
DE  LANGE,  among  other  observations  of  the  same  kind,  noted 
the  following  times  by  a  sidereal  chronometer,  when  the  moon's 

*  The  formula  (444)  is  essentially  the  same  as  that  given  by  OUDEMANS,  Astronom. 
Journal,  Vol.  IV.  p.  164.  The  method  itself  is  the  suggestion  of  Professor  KAISER 
of  the  Netherlands. 

f  Astronomical  Journal,  Vol.  IV.  p.  165. 


392  LONGITUDE. 

lower  limb  and  36  Capricorni  passed  the  same  fixed  horizontal 
threads : 

T  =  0*  38m  8'.62  T'=  0*  49-  53«.77 

The  difference  of  the  zenith  distances  indicated  by  the  level 
was 

I  — I'  =  4-  2".0 

The  chronometer  correction  was  A  T=  -\-  lm  3*.32,  and  the  rate 
in  the  interval  T'  —  T  was  insensible. 

The  assumed  latitude     was  <p  =    -  6°  9'  57".0 
"  longitude    "    L' =  —  7*  7-  37-.0 

We  have 

0  =  0*  39-  11».94  0'=  0*  50-  57'.09 

For  the  Greenwich  sid.  time  0  +  L'  =  17*  31wl  34*. 94, or  mean 
time  4/l  10'H  57*.00,  we  find,  from  the  Nautical  Almanac, 

a  =       21*  12-  5'.45  yl  =  4-  0'.0387 

d  =  —  20°  55'  8".9  ft  =  4-  0".1440 

*  =       57'  51".4  a'=       21*  20-  22'.45 

S  =       15'  47".8  d'=  —  22°  26'  30".5 

The  computation  by  (440)  gives 

C  =  52°  11'  49".44  C'=  53°  13'  57".30 

A  =  68°  14'.4  A'=  66°  30'.6 

j  =  81°18'.9  ?'=:80035'.2 

From  Table  III.  we  find 

9  —  j=  —  2'  27"  log  p  =  9.999983 

Since  the  same  fixed  threads  were  used  for  both  moon  and  star, 
we  have  m  =  m',  and  hence  also  sensibly  r  =  r' ;  therefore,  by 
(441),  we  find 

C  "  =  53°  13'  59".30  r  =  —  54".5  p  =  46'  21" .25 

C  —  :"==-•    62'    9".86  k  =  +  62'  9".17 

Hence,  by  (442), 

x  =  —  0".69  a  =  4-  0.5575  ±L  =  —  1-.24 

The  longitude  by  this  observation,  if  the  Ephemeris  is  correct, 
is  therefore 

i  ==  i'  +  Ai  =  —  7*  7"  38^.24 


BY    LUNAR    DISTANCES.  393 

If  we  compute  all  the  terms  of  (444),  we  shall  find 

A£  =  — 1'.24  —  24.84 Jo  —  0.27  W+ 23.84  dT—  24.24 8T' —  0.44 d±T 

+  24.28  cJa'-f  0.29  <W'+    1.79  *S  -f    1.44  ^     -  0.04  d<p 

This  shows  clearly  the  effect  of  each  source  of  error;  but  in  prac- 
tice it  will  usually  be  sufficient  to  compute  only  the  coefficients 
of  da  and  dd.  In  the  present  example,  therefore,  we  should  take 

A£  =  —  1-.24  —  24.84  da,  —  0.27  dd 

which  will  finally  be  fully  determined  when  8a  and  dd  have  been 
found  from  nearly  corresponding  observations  at  Greenwich  or 
elsewhere. 

SEVENTH   METHOD. — BY   LUNAR   DISTANCES. 

247.  The  distance  of  the  moon  from  a  star  may  be  employed 
in  the  same  manner  as  the  right  ascension  was  employed  in 
Arts.  229,  &c.,  to  determine  the  Greenwich  time,  and  hence  the 
longitude.  If  the  star  lies  directly  in  the  moon's  path,  the 
change  of  distance  will  be  even  more  rapid  than  the  change  of 
right  ascension  ;  and  therefore  if  the  distance  could  be  measured 
with  the  same  degree  of  accuracy  as  the  right  ascension,  it  would 
give  a  more  accurate  determination  of  the  Greenwich  time. 
The  distance,  however,  is  observed  with  a  sextant,  or  other  re- 
flecting instrument  (see  Vol.  II.),  which  being  usually  held  in 
the  hand  is  necessarily  of  small  dimensions  and  relatively  infe- 
rior accuracy.  Nevertheless,  this  method  is  of  the  greatest  im- 
portance to  the  travelling  astronomer,  and  especially  to  the 
navigator,  as  the  observation  is  not  only  extremely  simple  and 
requires  no  preparation,  but  may  be  practised  at  almost  any 
time  when  the  moon  is  visible. 

The  Ephemerides,  therefore,  give  the  true  distance  of  the 
centre  of  the  moon  from  the  sun,  from  the  brightest  planets,  and 
from  nine  bright  fixed  stars,  selected  in  the  path  of  the  moon, 
for  every  third  hour  of  mean  Greenwich  time.  The  planets  em- 
ployed are  Saturn,  Jupiter,  Mars,  and  Venus.  The  nine  stars, 
known  as  lunar-distance  stars,  are  a  Arietis,  a  Tauri  (Aldebarari), 
ft  Geminorum  (Pollux),  a  Leonis  (Regulus\  a  Virginis  (Spica), 
a  Scorpii  (Antares),  a  Aquilse  (Altair),  a  Piscis  Australia  (Fomal- 
haut],  and  a  Pegasi  (Markab}. 

The  distance  observed  is  that  of  the  moon's  bright  limb  from  a 


394  LONGITUDE. 

star,  from  the  estimated  centre  of  a  planet,  or  from  the  limh  of 
the  sun.  The  apparent  distance  of  the  moon's  centre  from  a  star 
or  planet  is  found  by  adding  or  subtracting  the  moon's  apparent 
(augmented)  semidiameter,  according  as  the  bright  limb  is  nearer 
to  or  farther  from  the  star  or  planet  than  the  centre.  The  ob- 
served distance  of  the  sun  and  moon  is  always  that  of  the  nearest 
limbs,  and  therefore  the  apparent  distance  of  the  centres  is  found 
by  adding  both  semidiameters.* 

The  apparent  distance  thus  found  differs  from  the  true  (geo- 
centric) distance,  in  consequence  of  the  parallax  and  refraction 
which  affect  the  altitudes  of  the  objects,  and  consequently  also 
the  distance.  The  true  distance  is  therefore  to  be  obtained  by 
computation,  the  general  principle  of  which  maybe  exhibited  in 
a  simple  manner  as  follows.  Let  Z,  Fig.  29, 
be  the  zenith  of  the  observer,  Mf  and  /S"  the  ob- 
served places  of  the  moon  and  star,  MM '  the 
parallax  and  refraction  of  the  moon,  SS'  the 
refraction  of  the  star,  so  that  M  and  S  are  the 
geocentric  places.  The  apparent  altitudes  of 
the  objects  may  either  be  measured  at  the  same 
time  as  the  distance,  or,  the  local  time  being 
known,  they  may  be  computed  (Art,  14).  The  apparent  zenith  dis- 
tances, and,  consequently,  also  the  true  zenith  distances,  are  there- 
fore known.  In  the  triangle  ZMfS'  there  are  known  the  three 
sides,  MfS'  the  apparent  distance  of  the  objects,  ZM'  the  apparent 
zenith  distance  of  the  moon,  and  ZSf  the  apparent  zenith  distance 
of  the  star;  from  which  the  angle  Z  is  computed.  Then,  in  the 
triangle  ZMS  there  are  known  the  sides,  ZM  the  moon's  true 
zenith  distance,  and  ZS  the  star's  true  zenith  distance,  and  the 
angle  Z;  from  which  the  required  true  distance  MS  is  computed. 
In  this  elementary  explanation  the  parallax  and  refraction  of 
the  moon  are  supposed  to  act  in  the  same  vertical  circle  ZM, 
whereas  parallax  acts  in  a  circle  drawn  through  the  moon  and 
the  geocentric  zenith  (Art.  81),  while  refraction  acts  in  the  vertical 
circle  drawn  through  the  astronomical  zenith.  Again,  when  the 
moon,  or  the  sun,  is  observed  at  an  altitude  less  than  50°,  it  is 
necessary  to  take  into  account  the  distortion  of  the  disc  produced 


*  We  may  also  observe  the  distance  from  the  limb  of  a  planet,  provided  the  sex- 
tant telescope  is  of  sufficient  power  to  give  the  planet  a  well-defined  disc ;  and  the 
planet's  semidiameter  is  then  also  to  be  added  or  subtracted. 


BY    LUNAR    DISTANCES.  395 

by  refraction  if  we  wish  to  compute  the  true  distance  to  the 
nearest  second  of  arc  (Art.  133).  These  features,  which  add 
very  materially  to  the  labor  of  computation,  cannot  be  over- 
looked in  any  complete  discussion  of  the  problem. 

Simple  as  the  problem  appears  when  stated  generally,  the 
strict  computation  of  it  is  by  no  means  brief;  and  its  importance 
and  the  frequency  of  its  application  at  sea,  where  long  computa- 
tions are  not  in  favor,  have  led  to  numerous  attempts  to  abridge 
it.  In  most  instances  the  abbreviations  have  been  made  at  the 
expense  of  precision  ;  but  in  the  methods  given  below  the  error 
in  the  computation  will  always  be  much  less  than  the  probable 
error  of  the  best  observation  with  reflecting  instruments:  so  that 
these  methods  are  entitled  to  be  considered  as  practically  perfect. 

With  the  single  exception  of  that  proposed  by  BESSEL,*  all  the 
solutions  depend  upon  the  two  triangles  of  Fig.  29,  and  may  be 
divided  into  two  classes,  ric/orous  and  approximative.  In  the 
rigorous  methods  the  true  distance  is  directly  deduced  by  the 
rigorous  formulae  of  Spherical  Trigonometry ;  but  in  the  approxi- 
mative methods  the  difference  between  the  apparent  and  the 
true  distance  is  deduced  either  by  successive  approximations  or 
from  a  development  in  series  of  which  the  smaller  terms  are 
neglected.  Practically,  the  latter  may  be  quite  as  correct  as  the 
former,  and,  indeed,  with  the  same  amount  of  labor,  more 
correct,  since  they  require  the  use  of  less  extended  tables  of 
logarithms.  I  propose  to  give  two  methods,  one  from  each  of 
these  classes. 

A. —  The  Rigorous  Method. 

248.  For  brevity,  I  shall  call  the  body  from  which  the  moon's 
distance  is  observed  the  sun,  for  our  formulae  will  be  the  eame 
for  a  planet,  and  for  a  fixed  star  they  will  require  no  other 
change  than  making  the  parallax  and  semidiameter  of  the  star 
zero. 


*  Astron.  Nach.  Vol.  X.  No.  218,  and  Astron.  Untersuchungcn,  Vol.  II.  BESSEL'S 
method  requires  a  different  form  of  lunar  Ephemeris  from  that  adopted  in  our 
Nautical  Almanacs.  But  even  with  the  Ephemeris  arranged  as  he  proposes,  the 
computation  is  not  so  brief  as  the  approximative  method  here  given,  and  its  supe- 
riority in  respect  of  precision  is  so  slight  as  to  give  it  no  important  practical 
advantage.  It  is,  however,  the  only  theoretically  exact  solution  that  has  been  given, 
and  might  still  come  into  use  if  the  measurement  of  the  distance  could  be  rendered 
much  more  precise  than  is  now  possible  with  instruments  of  reflection. 


396  LONGITUDE. 

Let  us  suppose  that  at  the  given  local  mean  time  T  the  obser- 
vation (or,  in  the  case  of  the  altitudes,  computation)  has  given 

d"=  the  apparent  distance  of  the  limbs  of  the  moon  and 

sun, 

A'=  the  apparent  altitude  of  the  moon's  centre, 
H'=  the  apparent  altitude  of  the  sun's  centre, 

and  that  in  order  to  compute  the  refraction  accurately  the 
barometer  and  thermometer  have  also  been  observed.  For  the 
Greenwich  time  corresponding  to  T9  which  will  be  found  with 
sufficient  accuracy  for  the  purpose  by  employing  the  supposed 
longitude,  take  from  the  Ephemeris 

s  =  the  moon's  semidiameter, 
S  =  the  sun's  " 

then,  putting 

d'=  the  apparent  distance  of  the  centres, 
s'  =  the  moon's  augmented  semidiameter, 

=  s  -f-  correction  of  Table  XII. 
we  have 

d'=d"  ±  s'±  S 

upper  signs  for  nearest  (inner)  limbs,  lower  signs  for  farthest 
(outer)  limbs. 

But  if  the  altitude  of  either  body  is  less  than  50°,  we  must 
take  into  account  the  elliptical  figure  of  the  disc  produced  by 
refraction.  For  this  purpose  we  must  employ,  instead  of  s'  and 
S,  those  semidiameters  which  lie  in  the  direction  of  the  lunar 
distance.  Putting 

q  =  ZM'S',  Q  =  ZS'M'   (Fig.  29) 

AS,  A$  —  the  contraction  of  the  vertical  semidiameters  of  the 
moon  and  sun  for  the  altitudes  h'  and  H'y 

the  required  inclined  semidiameters  will  be  (Art.  133) 
s'  —  AS  cos2  q  and  S  —  &S  cos2  Q 

The  angles  q  and  Q  will  be  found  from  the  three  sides  of  the 
triangle  ZM'S',  taking  for  df  its  approximate  value  d"  ±  s'  ±  S 
(which  is  sufficiently  exact  for  this  purpose,  as  great  precision  in 
q  and  Q  is  not  required),  and  for  the  other  sides  90°  —  h'  and 
90°  —  Hf.  If  we  put 

m  =  J(ft'+  JT'+  d') 


BY    LUNAR    DISTANCES.  397 

we  shall  have 

cos  m  sin  (m  —  H')  _  .       cos  m  sin  (m  —  A') 

sin2  J  0  = —A-  — - — i         sm2  }  Q= — ^  — — -L   (445) 

sm  d'  cos  h'  sin  6?'  cos  #' 

and  then  the  apparent  distance  by  the  formula 

d'=d"±(s'—  AS  cos2£)=h(£  —  A#cos2<))  (446) 

We  are  now  to  reduce  the  distance  to  the  centre  of  the  earth. 
We  shall  first  reduce  it  to  that  point  of  the  earth's  axis  which 
lies  in  the  vertical  line  of  the  observer.  Designating  this  point 
as  the  point  0,  Art.  97,  let 

dlt  hL,  Hl  =  the  distance  and  altitudes  reduced  to  the  point 

o, 

r,  R  =  the  refraction  for  the  altitudes  h'  and  H', 
x,  P  =  the  equatorial  hor.  parallax  of  the  moon  and 
sun. 

The  moon's  parallax  for  the  point  0  will  be  found  rigorously 
by  (127),  but  with  even  more  than  sufficient  precision  for  the 
present  problem  by  adding  to  ;r  the  correction  given  by  Table 
XIII.  Denoting  this  correction  by  ATT,  we  have 

7rt  =  TT  -j-  ATT 

hl  =  h'—r-\-iel  cos  (h'  —  r)       Hl  =  H'  —  E-\-P cos  (Hf  —  E)  (447) 

The  parallax  P  is  in  all  cases  so  small  that  its  reduction  to  the 
point  0  is  insignificant. 

If,  then,  in  Fig.  29,  M  and  S  represent  the  moon's  and  sun's 
places  reduced  to  the  point  0,  and  we  put 

Z  •=.  the  angle  at  the  zenith,  MZS, 

we  shall  have  given  in  the  triangle  M'ZSf  the  three  sides 
df,  90°  —  h',  90°  —  Hf,  whence 

'  cos  I  (hf  +H'  +  <Q  cos  \  (h'  +  H'-  d') 

COS    j  Z/  =  

cos  h'  cos  R' 

and,  then,  in  the  triangle  MZS  we  shall  have  given  the  angle  Z 
with  the  sides  90°  —  A,  and  90°  —  Hv  whence  the  side  MS  =  dv 
will  be  found  by  the  formula  [Sph.  Trig.  (17)], 

(^  -}-  HI)  —  cos  A,  cos  Hv  cos2  \  Z 


398  LONGITUDE. 

To  simplify  the  computation,  put 

m  =  I  (hf  -j-  H'  +  d') 
then  the  last  formula,  after  substituting  the  value  of  Z,  becomes, 

.  „  cos  7ii  cos  H] 

sm2  £  dl  =  cos2  2  (A!  -}-  Jfi) -  cos  m  cos  (m  —  a) 

cos  h'  cos  H' 

Let  the  auxiliary  angle  M  be  determined  by  the  equation 

cos  A,  cos  -ZT,     cos  m  cos  (m  —  d'} 

sm2  J/  =   ' •  -  (448) 

cos  h'  cos  H '      cos2  }  (A!  +  J5TO 

then  we  have* 

sin  $  dv  =  cos  £  (^  -f-  JE?i)  cos  Jf  (449) 

Finally,  to  reduce  the  distance  from  the  point  0  to  the  centre 
Fig.  so.          of  the  earth,  let  P  (Fig!  30)  be  the  north  pole  of 
the  heavens,  Ml  the  moon's  place  as  seen  from  the 
point  0,   M  the  moon's  geocentric  place,  8  the 
Mi    sun's  place  (which  is  sensibly  the  same  for  either 
"  >     point).      The  point   0  being   in  the  axis  of  the 
celestial  sphere,  the  points  M1  and  M  evidently  lie 
in  the  same  declination  circle   PMVM.      Hence, 
putting 

d  =  the  geocentric  distance  of  the  moon  and  sun  =  S M} 

d  =  the  moon's  geocentric  declination  =  90°  —  PM, 

dl  =  the  declination  reduced  to  the  point  0  =  90°  —  PMV 

A  =  the  sun's  declination  =  90°  —  PS, 


we  have,  in  the  triangles  PMS  and  MVMS, 

cos  d  —  cos  (<5,  -—  S)  cos  d        sin  J  —  sin  d  cos  d 

cos  PMS  =  -  -  =  

8jn  (^  —  <5)  gin  d  cos  d  sin  d 

WQ  may  put  cos  (dL  —  d)  =  1,  and,  therefore, 

sin  (3,  —  <T)   . 

cos  dl  —  cos  d  =  -  — -  (sin  A  —  sin  d  cos  d) 

cos  d 


*  This  transformation  of  the  formulae  is  due  to  BORDA,  Description  et  usage  du  cercle 
de  reflexion. 


BY   LUNAR   DISTANCES.  399 

and  since  d  —  dl  is  very  small,  we  may  put  cos  d^  —  cos  d  = 
sin  (d  —  d^  sin  dv  and  hence,  very  nearly, 


d.  —  d  t  sin  J          sin  d  \ 

d  —  dl  =  -1 I  — 

cos  d  \  sin  dl        tan  dl / 

Substituting  the  value  of  dl  —  d  from  (122), 

/  sin  J          sin  3  \ 

d  —  dl  —  A  TT  sin  <p  I  -  -  )  (450) 

\  sin  dl        tan  dv  ' 

in  which  <p  is  the  latitude  of  the  observer,  and  log  A  may  be 
taken  from  the  small  table  given  on  p.  116.  The  correction 
given  by  this  equation  being  added  to  dv  we  have  the  geocentric 
distance  d  according  to  the  observation. 

To  find  the  longitude,  we  have  now  only  to  find  the  Green- 
wich mean  time  T0  corresponding  to  d,  by  Art.  66,  and  then 

L=TQ-T  (451) 

EXAMPLE. — In  latitude  35°  K  and  assumed  longitude  150°  W., 
1856  March  9,  at. the  local  mean  time  T=  5/l  14™  6s,  the  ob- 
served altitudes  of  the  lower  limbs  and  the  observed  distance 
of  the  nearest  limbs  of  the  moon  and  sun  were  as  follows,  cor- 
rected for  error  of  the  sextant : 

h"  =  52°  34'  0"  H"  =  8°  56'  23"  d"  =  44°  36'  58".6 

The  height  of  the  barometer  was  29.5  inches,  Attached  therm. 
60°  F.,  External  therm.  58°  F. 

I  shall  put  down  nearly  all  the  figures  of  the  computation,  in 
order  to  compare  it  with  that  of  the  approximative  method  to  be 
given  in  the  next  article. 

1st.  The  approximate  Greenwich  mean  time  is  5h  14™  6*  -f  10* 
=  15A 14™  6*,  with  which  we  take  from  the  American  Ephemeris 

s  =  16'  23".l  *  =  60'  1".9  d  ==  -f  14°  19' 

£=16'    8".0  P=        8".6  J  =  -.    4°   3' 

2d.  To  find  the  apparent  semidiameters,  we  first  take  the 
augmentation  of  the  moon's  semidiameter  from  Table  XII., 
=  14".0,  and  hence  find 

sf  =  16'  37".  1 


400  LONGITUDE. 

Then  to  compute  the  contraction  produced  by  refraction  we  find 
from  the  refraction  table,  for  the  given  observed  altitudes,  the 

contractions  of  the  vertical  semidiameters  (Art.  132), 

AS  =  0".4  A£  =  9".6 

With  the  approximate  altitudes  and  distance  of  the  centres  we 
then  proceed  by  (445),  as  follows  : 

d'  =45°  10'  logcosecd'        '0.1493  log  cosec  d'        0.1493 

h'   =  52   51  log  sec  h'  0.2190 

//'  =    9   12  log  sec  H'          0.0056 

m    =  53   37  log  cos  m  9.7732  log  cos  m  9.7732 

m  —  H1  =  44  25  log  sin  (m  —  H'}    9.8450 

m  —  h'   =    0   46  _  log  sin  (m  —  A7)  8.1265 

9.9865  8.0546 

log  sin  %  q  9.9933  log  sin  £  Q          9.0273 

q=         159°  56'  Q=       12°  14' 

log  cos2  q  9.9456  log  cos2  Q  9.9800 

log  A*  9.6021  log  AS  0.9823 

9.5477  0.9623 

A*  cos2  q  —  0."4  AS  cos2  Q  =      9".2 

Hence  we  have,  by  (446), 


$'—  AS  cos2  ^  =        16  36  .7 

S—  AScos»C  =        15  58  .8 

d'=45     934.1 

3d.  To  find  the  apparent  and  true  altitudes  of  the  centres.  —  The 
apparent  altitudes  of  the  centres  will  be  found  by  adding  the 
contracted  vertical  semidiameters  to  the  observed  altitudes  of  the 
limbs.  The  apparent  altitudes,  however,  need  not  be  computed 
with  extreme  precision,  provided  that  the  differences  between 
them  and  the  true  altitudes  are  correct;  for  it  is  mainly  upon 
these  differences  that  the  difference  between  the  apparent  and 
true  distance  depends. 

The  reduction  of  the  moon's  horizontal  parallax  to  the  point 
0  for  the  latitude  35°  is,  by  Table  XIIL,  A;T  =  3".9;  and  hence 
we  have 

^  =  TT  -f  Ar  =  60'  5".8 

and  the  computation  of  the  altitudes  by  (447)  is  as  follows  : 


BY   LUNAR   DISTANCES.  401 

h"  =  52°  34'    0"  H"  =  8°  56'  23" 

Vert,  semid.  =         16  37  Vert,  semid.  =          15  58 


h'  =  52    50  37  H'  =  9    12  21 

Table  II.  r    =  42  .7  E    =         5  33  .6 

h'  —  r    =  52    49  54  .3  H'—  R    =  9      647.4 

log*,                   3.55700  logP                    0.9345 

log  cos  (A'  —  r)  9.78115  log  cos  (H'—  S)  9.9945 

3.33815          k  0.9290 

^  cos  (hf  —  r)    =         36'  18".5  P  cos  (W—  .R)  =              8".5 

^  =  ~53°  26'  12".8  HI=   9°  6'  55".9 

4th.  We  now  find  the  distance  rft  by  (448)  and  (449),  as  follows  : 

df  =  ±5°    9'34".l 

h'  =  52    50  37        log  sec      0.2189683 

H'  =    9    12  21        log  sec      0.0056300 

m  =  53    36  16~JL   log  cos      9.7733154 

m  —  d'  =    8    2642.     log  cos      9.9952654 

A!  =  53    26  12  .8   log  cos      9.7750333 

Jffj  =    9      6  55  .9   log  cos      9.9944803 

2)  9.7626927 
9.8813464 
J(A1+J1)  =  31    16  34  .4  log  cos       9.9318007    .........  £.9318007 

log  sin  M  9.9495457     log  cos  M  9.6583330 
}  d,  =  22    54     9  .2  log  sin  *  ^9^5901337 

dl  =  45    48  18  .4 

5th.   To  find  the   geocentric   distance,   we   have,   by   (450), 
for  <p  =  35°, 


log  A 
log* 
log  sin  (p 

7.8249 
3.5565 
9.7586 

8  =  _|_  140  19' 
J=-     4      3 

1  1400     .      . 

1  1400 

log  sin  A 
log  cosec  c 

wS.8490 
I,    0.1445 

log  sin  d      9.3932 
log  cot  di   9.9878 

nO.1335 

nO.5210 
—  3".3 

d  —  dl=—  4".7 

d  =  45°  48'  13".7 

6th.  To  find  the  Greenwich  mean  time  corresponding  to  d, 

VOL.  I.—  26 


402  LONGITUDE. 

and  hence  the  longitude,  according  to  Art.  66,  we  find  an  ap- 
proximate time  ( T)  -f-  t  by  simple  interpolation,  and  then  the 
required  time  TQ  =  (T)  +  t  +  A*,  taking  *l  from  Table  XX., 
with  the  arguments  /  and  A§  (=  increase  of  the  logarithms  in 
the  Ephemeris  in  3*),  as  follows : 

By  the  American  Ephemeris  of  1856  for  March  9,  we  have 

(r)=15»    0"    0*     (//):=  45°  40' 54"  §  =  0.2510     A §=  +  17 

d    =45    48  13  .7 

t    =    0   13      4  7  19  .7     log   =  2.6432 

A£=_          _1  log  t  =  2.8942 

Tn=  15  13  3 
T=  5  14  6 
L  =  9  58  57 

B. —  The  Approximative  Method. 

249.  I  shall  here  give  my  own  method  (first  published  in  the 
Astronomical  Journal,  Vol.  II.),  as  it  yet  appears  to  me  to  be 
the  shortest  and  most  simple  of  the  approximative  methods 
when  these  are  rendered  sufficiently  accurate  by  the  introduction  of  all 
the  necessary  corrections.  Its  value  must  be  decided  by  the  im- 
portance attached  to  a  precise  result.  There  are  briefer  methods 
to  be  found  in  every  work  on  Navigation,  which  will  (and  should) 
be  preferred  in  cases  where  only  a  rude  approximation  to  the 
longitude  is  required. 

As  before,  let 

h',  Hf  =  the  apparent  altitudes  of  the  centres  of  the  moon 

and  sun, 

d"  =  the  observed  distance  of  the  limbs, 
5,  S  =  their  geocentric  semidiameters, 
TT,  P  =  their  equatorial  horizontal  parallaxes, 

sf  =  the  moon's  semidiameter,  augmented  by  Table 

XII., 
TTj  =  the  moon's  parallax,  augmented  by  Table  XIII. 

We  shall  here  also  first  reduce  the  distance  to  the  point  0  of 
Art.  97.  The  contractions  of  the  semidiameters  produced  by 
refraction  will  be  at  first  disregarded,  and  a  correction  on  that 
account  will  be  subsequently  investigated.  If  then  in  Fig.  29, 
p.  394,  Mf  and  Sf  denote  the  apparent  places,  M and  S  the  places 
reduced  to  the  point  0,  we  shall  here  have 


BY    LUNAR    DISTANCES.  403 

d'  =  d"±J±S  =  M'S',  d,  =  MS, 

V  =  90°  —  ZM' ,  H '  s=  90°  —  ZS', 

ht  =  90°  —  ZM,  If,  =  90°  —  ZS, 

and  the  two  triangles  give 

cos  dl  —  sin  A,  sin  HI       cos  d'  —  sin  h'  sin  H1 
cosZ=- -== — 

COS  A!  COS  Hi  COS  h  COS  H 

from  which,  if  we  put 

sin  A.  sin  H.  cos  A.  cos  H 

Tl\ •  71  

sin  A'  sin  H'  cos  A'  cos  H' 

we  derive 

cos  d'  —  cos  dl  =  (1  —  n)  cos  d'  -|-  (n  —  m)  sin  A'  sin  H'        (<z) 
Put 


then  we  have 

cos  d'  —  cos  dl  =  2  sin  £  Ad  sin  (d'  -f  £  *d)  (c) 

and 

COS  (A'  H-  A^     COS  (H'—  Aff) 


COS  /l  COS 


\      / 

r\ 


_n(A;4-  ^AA)        2  sin  ^  A  g  sin  (Hr— 


cos  h'  cos  If' 

4  sin  j  Afr  sin  j  AlTsin  (hf  -f  ^  A/t)  sin  (//'  —  ^  Aff) 
cos  h'  cos  If  ' 

Also 

sin  h'  cos  A,  sin  JET  cos  If,  —  cos  h'  sin  A,  cos  jff'  sin  If, 

n  —  m  —  -  1  -  1  —  .  -  i  -  1 

sin  A'  cos  A'  sin  If'  cos  If  ' 

substituting  in  which  the  values 

2  sin  A'  cos  At  =  sin  (2  A'   -f-  AA)   --  sin  A  A 
2  cos  A'  sin  Af  —  sin  (2  A'    -|-  AA)    -f-  sin  AA 
2  sin  H  '  cos  7^=  sin  (2  If'—  A#)  +  sinAlf 
2  cos  if  'sin  J^  =  sin  (2  IT  —  A^f)  —  sin  &H 
we  find 

_sinAgsin(2A'-|-AA)  —  sinAAsin(2J?"'—  Ag) 
n~~m-  2  sin  A'  cos  A'  sing'  cos  #' 


404  LONGITUDE. 

Substituting  (c),  (c/),  and  (e)  in  (a\  and  at  the  same  time,  for 
brevity,  putting 

2  sin  J  A/I  sin  (A'  -f  £  AA) 
1==  cos  A' 

_  sin  AA  sin  (2  If'  —  AJT) 
1  ~  2  cos  A'  cos  If' 

2  sin  \  AlT  sin  (JET  —  i  AJ?T) 

C,  = 


A  = 


cos  If' 

sin  &H  sin  (2  A'  -f  AA) 


2  cos  A'  cos 
have 


This  formula  is  rigorously  exact;  but,  since  AC?  is  always  less 
than  1°,  it  will  not  produce  an  error  of  0/x.l  to  substitute  the  arcs 
J  A^,  J  A^,  &c.  for  their  sines,  or  J  AC/  sin  1",  J  A/i  sin  1",  &c.  for 
sin  J  AC/,  sin  J  A/i,  &c.  ;  and  therefore  we  may  write 


in  which  J.j,  ^,  Ci,  -D1?  now  have  the  following  signification  : 

4,=       -^-.  sin  (hf  +£  A/0 
cos  A' 

_  AA      Bin(2gy— 

1  ~         cos  h'  '          2  cos  I 


A/f      8in(2A'+  AA) 
1-         008^  '    ~¥cosA' 

The  next  step  in  our  transformation  consists  in  finding  con- 
venient and  at  the  same  time  sufficiently  accurate  expressions 
of  A/<  and  A^f.  Let 


r,  R  —  the  true  refractions  for  the  apparent  altitudes  h!  and 
H'; 

then  we  have,  within  less  than  0".l, 

AA  =  TTI  cos  (A'  —  r)  —  r 


BY    LUNAR    DISTANCES.  405 

If  we  neglect  r  in  the  term  ^  cos  (A' —  r),  the  error  in  this  term 
will  never  exceed  1" ;  but  even  this  error  will  .be  avoided  by 
taking  the  approximate  expression 

cos  (A'  —  f)  =  cos  h'  -j-  sin  r  sin  A' 
and  we  shall  then  have 

AA  =  7^  cos  A'  —  r  -f  -!  sin  r  sin  h' 

/          rr,  sin  r  sin  A'  \ 
=  (rlCosA'— r)    1  +  - 

\  ^  cos  A'  —  r  / 

Since  the  second  term  of  the  second  factor  produces  but  1" 
in  AA,  we  may  employ  for  it  an  approximate  value,  which  will 
still  give  AA  with  great  precision.  Denoting  this  term  by  A,  we 

have 

TTj  sin  r  sin  A'         sin  r  tan  A' 

7rx  cos  A'  —  r        i  r 

TTj  COS  k' 

or,  very  nearly, 

A  =  sinr  tan  A'(l  -] \ 

KI  cos  A'  / 

If  we  put 

r  =  a  cot  A', 

in  which  a  has  the  value  given  in  Table  II.,  we  have 
A=asinl"/l  -f 


Kow,  a  increases  with  A',  but  in  such  a  ratio  that  A:  remains  very 
nearly  constant  for  a  constant  value  of  7rr  We  may  without 
sensible  error  take  ^  —  57'  30"  —  3450",  which  is  about  the 
mean  value  of  rrv  and  we  shall  find  for  a  mean  state  of  the  air, 
by  the  values  of  a  given  in  Table  II., 

forA'=    5°  k  =  0.000291 

A'  =45  Jc  =  0.000286 

A'  =90  A  =  0.000285 

Hence,  if  we  take 

k  =  0.00029 
the  formula 

AA  =  (*!  cos  A'  —  r)  (1  +  A)  (452) 


will  give  AA  within  ^QM  °f  its  whole  amount,  that  is,  within  less 
than  0".02  in  a  mean  state  of  the  air.     For  extreme  variations 


406  LONGITUDE. 

of  the  density  of  the  air,  it  is  possible  that  the  refraction  may 
be  increased  by  its  one-sixth  part,  and  k  will  also  be  increased 
by  its  one-sixth  part.  But,  as  the  term  depending  on  k  is  not 
more  than  1",  the  error  in  A/J,  even  in  the  improbable  case 
supposed,  will  not  be  greater  than  0".16.  The  formula  (452) 
may  therefore  be  regarded  as  practically  exact  with  the  value 
k  =  0.00029. 

A  strict  computation  of  the  sun's  or  a  planet's  altitude  requires 

the  formula 

=R  —  P  cos  (Hr—  R) 


but  P  is  in  all  cases  so  small  that  the  formula 

A  H  =  E  —  P  cos  11'  (453) 

will  always  be  correct  within  a  very  small  fraction  of  a  second. 
Now,  let 

*'=  (454) 


The  quantities  rf  and  K'  computed  from  the  mean  values  of  the 
refraction  are  given  in  Table  XIV.  under  the  name  "  Mean 
Reduced  Refraction  for  Lunars."  The  numbers  of  the  table 
are  corrected  for  the  height  of  the  barometer  and  thermometer 
by  means  of  Table  XIV.  A  and  B.  These  tables  are  computed 
from  BESSEL'S  refraction  table,  assuming  the  attached  ther- 
mometer of  the  barometer,  and  the  external  thermometer,  to 
indicate  the  same  temperature,  which  is  allowable  in  our  present 
problem.*  By  the  introduction  of  r'  and  R  ',  we  obtain 


cos  hf 
and  the  coefficients  of  formula  (g)  become 


*  If  it  is  desired  to  compute  /•'  and  R'  with  the  utmost  rigor,  it  can  be  done  by 
Table  II.,  by  taking  (Art.  107) 


sin  h'  sin  //' 

The  tables  XIV.  and  XIV.  A  and  B  give  the  correct  values  to  the  nearest  second  in  all 
practical  cases. 


BY   LUNAR   DISTANCES.  40T 

A,  =       (^  —  r')  (1  +  A)  sin  (hf  +  J  AA) 


2  cos  h' 

The  term  J^  Cj  sin  1"  cosd'  is  very  small,  its  maximum  being 
only  1".  It  is  easy  to  obtain  an  approximate  expression  for  it 
and  to  combine  it  with  the  term  Alcosdf.  In  so  small  a  term 
we  may  take 

d  sin  V  =  —  R'  sin  1"  sin  H'  =  —  sin  E  tan  H'  =  —  k 
and  hence 

Al  —  A&  sin  l"  =  Al  (1  +  k)  =  (^  —  r')  (1  +  A')2  sin  (A'  +  }  *A) 

If  now  we  put 

^  =  (1+  ft 


sin  A' 


B=(l+K) 


sn 


_ 


sin  2  A' 


and 


A'  —       (X  —  r')  A  sin  A'  cot  d' 
B'  =  —  (7rt  —  r')  -B  sin  IT'  cosec  dr 
C'=  —  (R'—P}  C  sin  H'  cot  d' 
D'=       (Rf— P)Dsin  A' cosec  d' 

the  formula  (#)  becomes,  when  divided  by  sin  d' ', 


(455) 


0456) 


sin  d' 

the  first  member  of  which  may  be  put  under  the  form 
/          2  sin  U^  cos  (<i'+ 

Aal   1   -|- ; — 

\  sm  d' 


408  LONGITUDE. 

so  that  if  we  put 


in  l"cos(<r-f 


or,  within  0".15, 

x  =  —  £  Ad2  sin  I"  cot  d'  (457) 

we  have 

*d  =  A'  +  J5'  +  C"  +  D''  -|-  x  (458) 

The  terms  -4/,  -B',  C",  and  Dr  are  computed  directly  from  the 
apparent  distance  and  altitudes  by  (456),  and  with  sufficient 
accuracy  with  four-figure  logarithms.  The  logarithms  of  A,  B,  C,  D, 
are  given  in  Table  XV.,  log  A  and  log  D  with  the  arguments 
TTI  —  rf  and  hf  ;  log  B  and  log  C  with  the  arguments  Rr  —  P 
and  Hf.  In  the  construction  of  this  table  A  A  and  &H  are  com- 
puted by  (452)  and  (453),  and  then  the  logarithms  of  A,  B,  C,  D, 
by  (455). 

The  sum  A'+B'+C'  +  Df  is  called  the  "  first  correction  of  the 
distance,"  and,  being  very  nearly  equal  to  A<i,  is  used  as  the  argu- 
ment of  Table  XVI.  ,  which  gives  #,  or  the  "  second  correction 
of  the  distance,"  computed  by  (457).  When  x  is  greater  than  30" 
and  the  distance  small,  it  will  be  necessary  to  enter  this  table  a 
second  time  with  the  more  correct  value  of  &d  found  by  em- 
ploying the  first  value  of  x. 

The  correction  AC/  being  thus  found  and  added  to  d',  we  have 
dv  or  the  distance  reduced  to  the  point  0.  The  reduction  to  the 
centre  of  the  earth  is  then  made  by  (450).  This  reduction  is 
also  facilitated  by  a  table.  If  we  put 

/  sin  A         sin  d  \ 
N  =  A-   - 

\  sm  dl       tan  dl  / 

and  then 

sin  8  sin  A 

a  —  —  A-K  -  o  =  Ax  —  - 

tan  dv  sin  dl 

we  shall  have 

!T=a  +  b  (459) 

and  a  and  b  can  be  taken  from  Table  XIX.  where  a  is  called  "  the 
first  part  of  N9"  and  b  "  the  second  part  of  N."     We  then  have 

d  _^:=  ATgin  ?>  (460) 

which  is  the  correction  to  be  added  to  dl  to  obtain  the  geocentric 
distance  d.     Table  XIX.  is  computed  with  the  mean  value  oi 


BY    LUNAR    DISTANCES.  409 

it  =  57'  30",  which  will  not  produce  more  than  1"  error  in 
d  —  dv  in  any  case.  But,  if  we  wish  to  compute  the  correction 
for  the  actual  parallax,  we  shall  have,  after  finding  N  by  the 
table, 

d  —  d  _  ==  N  sin  <p  X  —  —  (460*) 

3450" 

n  being  in  seconds. 

The  trouble  of  finding  the  declinations  of  the  bodies  and  the 
use  of  Table  XIX.  would  be  saved  if  the  Almanac  contained  the 
logarithm  of  N  in  connection  with  the  lunar  Ephemeris.  The 
value  of  log  N  in  the  Almanac  would,  of  course,  be  computed 
with  the  actual  parallax,  and  (460)  would  be  perfectly  exact. 

We  have  yet  to  introduce  corrections  for  the  elliptical  figure 
of  the  discs  of  the  moon  and  sun  produced  by  refraction.  These 
corrections  are  obtained  by  Tables  XVII.  and  XVIII.,  which  are 
constructed  upon  the  following  principles.  Let 

ASj,  A/Sj  —  the  contractions  of  the  vertical  semidiameters, 
AS,  A$  =  the  contractions  of  the  inclined  semidiameters; 

then  we  have  (Art,  133) 

AS  =  ASj  cos2  q  A£  =  AA^  cos2  Q 

where  q  =  the  angle  ZM'S'  (Fig.  29)  and  Q  =  ZS'M'.  We 
have 

sin  H'  —  sin  hf  cos  dr 
cos  a  =.  -       - 
cos  h!  sin  d' 

But,  by  (456), 


sin  H'  B'  sin  h'  cos  d 


cos  h'  sin  d'  B  (-x  —  rf)  cos  h'         cos  h'  Bind'       A  (^  —  r')  cos  h' 

so  that 


If  we  put  A  —  1  and  B  —  1,  which  are  approximate  values,  we 
shall  have 

A'      Br 


(401) 


410  LONGITUDE. 

In  order  to  ascertain  the  degree  of  accuracy  of  this  formula, 
we  observe  that  the  errors  in  cos  q  produced  by  the  assumption 
A  =  1,  B  =  1,  are 

BiDjr 


tand'  cos  ft'  Bind' 

the  errors  in  c.os2<?  are 

2ecos  q  2^  cos  q 

and  the  errors  in  A  s  are,  therefore, 


-  "  1)  tan  A'  cos  g  ,_  2AS,  (1  —  J5)  sin  IT  cos  g 

tan  d'  cos  A'  sin  d' 

In  order  to  represent  extreme  cases,  let  us  suppose  q  =  0  and 
H'=  90°,  which  will  give  el  and  e^  their  greatest  values;  then 
we  shall  find  for  the  different  values  of  hf  the  following  errors  : 


A' 

fjtan  d' 

ej'  sin  c 

5° 

0".45 

0".02 

10 

.16 

.00 

15 

.08 

.00 

80 

.02 

.00 

50 

.00 

.00 

It  can  only  be  for  very  small  values  of  d'  that  the  error  el  can  be 
important,  even  for  k'  =  5°  ;  and,  as  these  small  values  of  the 
distance  are  always  avoided  in  practice,  our  formula  (461)  may 
be  considered  quite  perfect. 

In  the  same  manner,  we  shall  find 

[C'4-Df        ~l2 
(*•-£«»*»]     ,       (462> 

which  is  even  more  accurate  than  (461). 

These  formulae  are  put  into  tables  as  follows.  For  the  moon, 
Table  XVII.  A,  with  the  arguments  h1  and  r^—  r'  ,  gives  the 
value  of 


where  /  is  an  arbitrary  factor  (=  18000000)  employed  to  give  g 
convenient  integral  values.  Then  Table  XVH.B,  with  the  argu- 
ments g  and  Af  -\-  Bf,  gives 


BY   LUNAR    DISTANCES.  411 


For  the  sun,  Table  XVIII.  A,  with  the  arguments  H'  and  E'  —  P, 
gives  the  value  of 


in  which  F=  -TTT-;  and  Table  XVIILB  gives 


In  these  tables  J/-f-  -B'  is  called  the  "whole  correction  of  the 
moon,"  and  C'-{-Df  the  "whole  correction  of  the  sun."  As 
these  quantities  are  furnished  by  the  previous  computation  of 
the  true  distance,  the  required  corrections  are  taken  from  the 
tables  without  any  additional  computation. 

The  values  of  AS  and  A$  are  applied  to  the  distance  as  follows  : 
when  the  limb  of  the  moon,  nearest  to  the  star  or  planet  is 
observed,  AS  is  to  be  subtracted,  and  when  the  farthest  limb  is 
observed,  AS  is  to  be  added  ;  when  the  sun  is  observed,  both  AS 
and  A$  are  to  be  subtracted  from  d. 

In  strictness,  these  corrections  should  be  applied  to  the  dis- 
tance d',  and  the  distance  thus  corrected  should  be  employed  in 
computing  the  values  of  A',  Br,  C',  and  D'.  This  would 
require  a  repetition  of  the  computation  after  AS  and  A$  had  been 
found  by  a  first  computation  ;  but  this  repetition  will  rarely 
change  the  result  by  0".5.  In  the  extreme  and  improbable  case 
when  the  distance  is  only  20°  and  one  body  is  at  the  altitude  5° 
and  the  other  directly  above  it  in  the  same  vertical  circle  (so  that 
the  entire  contraction  of  the  vertical  semidiameter  comes  into 
account),  such  a  repetition  would  change  the  result  only  1".8  ; 
and  even  this  error  is  much  less  than  the  probable  error  of 
sextant  observations  at  this  small  altitude,  where  the  sun  and 
moon  already  cease  to  present  perfectly  defined  discs. 

250.  I  shall  now  recapitulate  the  steps  of  this  method. 
1st.  The  local  mean  time  of  the  observation  being  7}  and  the 
assumed  longitude  X,  take  from  the  Ephemeris,  for  the  approxi- 


412 


LONGITUDE. 


mate  Greenwich  time  T  -f  Z/,  the  quantities  s,  S,  n,  P,  £,  and  A. 
(For  the  sun  we  may  always  take  P  =  8".  5  ;  for  a  star,  S  =  0, 
P=0.) 

2d.  If  h",  H",  d"  denote  the  observed  altitudes  and  distance 
of  the  limbs,  find 

.   s'  =  s  -j-  correction  of  Table  XII., 
Tfl=  T  -j-  correction  of  Table  XIII., 

and  the  apparent  altitudes  and  distance  of  the  centres, 

h'=h"+J,  H'=H"+S,  d'=d"±s'±S 

upper  signs  for  upper  and  nearest  limbs,  lower  signs  for  lower 
and  farthest  limbs. 

For  the  altitudes  h'  and  77',  take  the  "  reduced  refractions" 
r'  and  R  from  Table  XIV.,  correcting  them  by  Table  XIV.A 
and  B  for  the  barometer  and  thermometer.  Then  compute  the 
quantities 


A'  = 


*,—  r')  Asin  h'cotd' 


C'  =  —  (Rf—  P)  Csin  H'eoid' 
D'=       M' 


for  which  the  logarithms  of  A,  B,  C,  and  D  are  taken  from 
Table  XV.  In  this  table  the  argument  ^  —  r'  is  called  the 
"  reduced  parallax  and  refraction  of  the  moon,"  and  R'  —  P  the 
"reduced  refraction  and  parallax  of  the  sun  (or  planet)  or  star." 
For  a  star  this  argument  is  simply  R. 

When  d'>  90°,  the  signs  of  A'  and  C'  will  be  reversed.  It 
may  be  convenient  for  the  computer  to  determine  the  signs  by 
referring  to  the  following  table  : 


A' 

B' 

C" 

D' 

d1  <  90° 

+ 

— 

— 

+ 

<f  >90° 

— 

— 

+ 

-f 

3d.  The  terms  A'  and  Br,  which  depend  upon  the  moon's 
parallax  and  refraction,  may  be  called  the  first  and  second  parts 
of  the  moon's  correction,  and  the  sum  A'  -f-  Bf  the  "  whole  cor- 
rection of  the  moon."  In  like  manner,  C'  and  D'  may  be  called 
the  first  and  second  parts  of  the  sun's,  planet's,  or  star's  correc- 


BY   LUNAR    DISTANCES.  •        413 

tion,  and  the  sura  Cf  +  Df  the  "  whole  correction  of  the  sun, 
planet,  or  star." 

The  sum  of  these  corrections  —  A'  -f-  B*  -f-  C"  +  Df  may  he 
called  the  "first  correction  of  the  distance."  Taking  it  as  the 
upper  argument  in  Tahle  XVI. ,  find  the  second  correction  =  x, 
the  sign  of  which  is  indicated  in  the  tahle. 

4th.  Take  from  Tahle  XVII. A  and  B  the  contraction  of  its 
inclined  semidiametcr  =  AS.  If  the  sun  is  the  other  body,  take 
also  the  contraction  from  Tahle  XVIII. A  and  B,  =  A&  The 
sign  of  either  of  these  corrections  will  be  positive  when  the 
farthest  limb  is  observed,  and  negative  when  the  nearest  limb  is 
observed. 

5th.  The  correction  for  the  compression  of  the  earth  is  = 
.ZVsin  <p,  <p  being  the  latitude;  and  N  may  be  accurately  com- 
puted by  the  formula 


sin  dl       tan  dl 

or  it  may  be  found  within  V  by  Table  XIX.,  the  mode  of  con- 
sulting which  is  evident.  The  sign  of  A7" sin  tp  will  be  determined 
by  the  signs  of  jY  and  sin  tp,  remembering  that  for  south  latitudes 
sin  tp  is  negative. 

All  the  corrections  being  applied  to  d',  we  have  the  geocen- 
tric distance  d;  and  hence  the  corresponding  Greenwich  time 
and  the  longitude. 

EXAMPLE. — Let  us  take  the  example  of  the  preceding  article 
(p.  399),  in  which  the  observation  gives 

1856,  March  9th,  $  =  35°. 

T       =    5*  14"  6-        }/i"=52°34'   0"        Barom.29.5in. 
Assumed  £       =  10_JM)         Qff"=    8    5623          Therm.  58°  F. 
Approx.  Gr.  T.  =  15~TT~6    Di  iO  d"  =44    36  58.6 

By  the  Ephemeris,  we  have 

s  =  16'  23".l  TT  ==  60'  1".9     #=16'  8".0    P=  8".6 

Table  XII.     -f  14  .0    Tab.  XIII.        -f  3  .9     d  =  -f  14°     J  =  —  4° 

sf  =  i (HrT T  ff t  =  60  5 .8 

The  computation  may  be  arranged  as  follows: 


414 


LONGITUDE. 


A"=  52°34'.0 
«'=  4-  16.6 
h'=  62  50.6 


Table  XIV. 

"       "     A. 
"       "     B. 


r'  = 


Ill  .1 

60    5  .8 

58  54  .7 


Cf         

//•  = 

8°  56'.4 
16.1 

S  — 

1°  36'  58".6 
16  37  .1 
16     8  .0 

9    12.5 

5' 49".  6 
—    6  . 
-    6  . 

R'=      5  37  .6 
P=  8  .6 

R'—P=      529  .0 


d'  =  45     9  43  .7 


(Table  XV.)  log  A 

log  K  —  »" 
log  sin  h' 
log  cot  c?' 

log  A' 


0.0019 
3.5484 
9.9015 
9.9975 

3.4493 


(Table  XV.)  log  B  9.9981 

log  (^  -  r')  3.5484 

log  sin  //'  9.2042 

log  cosec  df  0.1493 
logJ?' 


w2.9000 
#'  =  — 13'14".3 
A'-\-B'=  +3339  .6 

Table  XIX.  1st  Part  of  N=  —  6' 


(Table  XV.)  log  C  9.9978 

log  (R'  —  P)  2.5172 

log  sin  //'  9.2042 

log  cot  d'  9.9975 

log  C'  nl.7167 

C"=—  52".  1 


(Table  XV.)  log  D 
log  (72'—  P) 
log  sin  h' 
log  cosec  c?' 

log     D' 

J9'=  +6'  8".  7 
'.=     -516  .6 


9.9987 
2.5172 
9.9015 
0.1493 
2.5667 


•=  _  8.     6=  35°. 


1st  corr.  —  +  38'  66". 2 
(Table  XVI.)     2d  corr.  =  13  .5 

(Table  XVII.)  A*  =  0  . 

(Table  XVIII.)  £,S^=  -  9  . 

^—          4  .6 


=  4648  12  .8 


This  result  agrees  with  that  found  by  the  rigorous  method  on 
p.  401,  within  1". 

To  find  the  longitude,  we  now  have,  by  the  American  Eplie- 
mcris  for  March  9, 


=  15»    0-  0* 


t    = 
Table  XX. 


0   13     3 


T  =    5 


13 
14 


=    9   58  56 


(d)  =  45°  40'  54" 
d   =  45  48  13 
7  19 


Q=  0.2510     J§= 

log    =  2.6425 
log  t=  2.8935 


BY   LUNAR    DISTANCES.  415 

251.  Iii  consequence  of  the  neglect  of  the  fractions  of  a  second 
in  several  parts  of  the  above  method,  it  is  possible  that  the  computed 
distance  may  be  in  error  several  seconds,  but  it  is  easily  seen 
that  the  error  from  this  cause  will  be  most  sensible  in  cases 
where  the  distance  is  small ;  and,  since  the  lunar  distances  are 
given  in  the  Ephemeris  for  a  number  of  objects,  the  observer 
can  rarely  be  obliged  to  employ  a  small  distance.     If  he  confines 
himself  to  distances  greater  than  45°  (as  he  may  readily  do),  the 
method  will  rarely  be  in  error  so  much  as  2",  especially  if  he 
also  avoids  altitudes  less  than  10°.     When  we  remember  that 
the  least  count  of  the  sextant  reading  is  10",  and  that  to  the 
probable  error  of  observation  we  must  add  the  errors  of  gradua- 
tion, of  eccentricity,  and  of  the  index  correction,  it  must  be  con- 
ceded that  we  cannot  hope  to  reduce  the  probable  error  of  an 
observed  distance  below  5",  if  indeed  we  can  reduce  it  below 
10".     Our  approximate  method  is,  therefore,  for  all  practical 
purposes,  a  perfect  method,  in  relation  to  our  present  means  of 
observation. 

252.  If  the  altitudes  have  not  been  observed,  they  may  be 
computed  from  the  hour  angles  and  declinations  of  the  bodies, 
the  hour  angles  being  found  from  the  local  time  and  the  right 
ascensions.    But  the  declination  and  right  ascension  of  the  moon 
will  be  taken  from  the  Ephemeris  for  the  approximate  Green- 
wich time  found  with  the  assumed  longitude.  If,  then,  the  assumed 
longitude  is  greatly  in  error,  a  repetition  of  the  computation  may 
be  necessary,  starting  from  the  Greenwich  time  furnished  by  the 
first.     As  a  practical  rule,  we  may  be  satisfied  with  the  first 
computation  when  the  error  in  the  assumed  longitude  is  not 
more  than  30*.     In  the  determination  of  the  longitude  of  a  fixed 
point  on  land,  it  will  be  advisable  to  omit  the  observation  of  the 
altitudes,  as  thereby  the  observer  gains  time  to  multiply  the 
observations  of  the  distance.     But  at  sea,  where  an  immediate 
result  is  required  with  the  least  expenditure  of  figures,  the  alti- 
tudes should  be  observed. 

253.  At  sea,  the  observation  is  noted  by  a  chronometer  regu- 
lated to  Greenwich  time,  and  the  most  direct  employment  of  the 
resulting  Greenwich  time  will  then  be  to  determine  the  true 
correction  of  the  chronometer.     This  proceeding  has  the  advan- 


416  LONGITUDE. 

tage  of  not  requiring  an  exact  determination  of  the  local  time  at 
the  instant  of  the  observation. 

For  example,  suppose  the  observation  in  the  example  above 
computed  had  been  noted  by  a  Greenwich  mean  time  chrono- 
meter which  gave  157t  10™  0',  and  was  supposed  to  be  slow  4W  6". 
The  true  Greenwich  time  according  to  the  lunar  observation 
was  15*  13'"  0s,  and  hence  the  true  correction  was  -f  3W  0".  "With 
this  correction  we  may  at  any  convenient  time  afterwards  deter- 
mine the  longitude  by  the  chronometer  (Art.  214). 

In  this  way  the  navigator  may  from  time  to  time  during  a 
voyage  determine  the  correction  of  the  chronometer,  and,  by 
taking  the  mean  of  all  his  results,  obtain  a  very  reliable  correc- 
tion to  be  used  when  approaching  the  land.  He  may  even 
determine  the  rate  of  the  chronometer  with  considerable  accu- 
racy by  comparing  the  mean  of  a  number  of  observations  in 
the  first  part  of  the  voyage  with  a  similar  mean  in  the  latter 
part  of  it. 

254.  To  correct  the  longitude  found  by  a  lunar  distance  for  errors 
of  the  Ephemeris. — In  relation  to  the  degree  of  accuracy  of  the 
observation,  we  may  in  the  present  state  of  the  Ephemeris  regard 
all  its  errors  as  insensible  except  those  which  affect  the  moon's 
place.  If,  therefore,  the  longitude  of  a  fixed  point  has  been 
found  by  a  lunar  distance  on  a  certain  date,  the  corrections  of 
the  moon's  right  ascension  and  declination  are  first  to  be  found 
for  that  date  from  the  observations  at  one  or  more  of  the  prin- 
cipal observatories,  and  then  the  correction  of  the  longitude  will 
be  found  as  follows.  Let 

a,  8  =  the  right  ascension  and  declination  of  the  moon  given 

in  the  Ephemeris  for  the  date  of  the  observation, 
A,  J  =  those  of  the  sun,  planet,  or  star, 
da,  <5<5=  the  corrections  of  the  moon's  right  ascension  and 

declination, 

dd  =  the  corresponding  correction  of  the  lunar  distance, 
SL  =  the  corresponding  correction  of  the  computed  longi- 
tude; 

In  Fig.  30,  M  and  S  being  the  geocentric  places  of  the  two 
bodies,  as  given  in  the  Ephemeris,  and  d  denoting  the  distance 
MS,  we  have 

cos  d  —  sin  d  sin  J  -~  cos  ^  cos  ^  co8    a  —  ^ 


BY   LUNAK   DISTANCES.  417 

by  differentiating  which  we  find 

cos  d  cos  A  sin  (a  —  A)  , 

3d  =  -       —  ^  -  £,  .  «Ja 

sin  d 
cos  d  sin  A  —  sin  d  cos  A  cos  (a  —  A] 

sin  d 
If  then 

v  =  the  change  of  distance  in  3*, 
we  shall  have 

9L  =  —  dd  X  |*  (465) 

in  computing  which  we  employ  the  proportional  logarithm  of  the 

3* 
Ephemeris,  Q  =  log  —  ,  reduced  to  the  time  of  the  observation. 

EXAMPLE.  —  At  the  time  of  the  observation  computed  in  Art,, 
250,  we  have 

Moon,  a  =    2*  11-  14'  d  =  +  14°  18'.4 

Sun,     A  =  23   22    25  J=-     4      3.1 

a  —  A  =    2   49    19  d  =       45    48  .2 
==    42°  19'.8 

with  which  we  find,  by  (464), 

dd  =  0.908  da  -f  0.350  S3 
and  hence,  by  (465),  with  log  Q  =  0.2511, 

dL  =  —  1.62  da  —  0.62  38 

Suppose  then  we  find  from  the  Greenwich  observations  da  = 
-  0*.38  =     -  5".7  and  dd  =  —  4".0,  the  correction  of  the  longi- 
tude above  found  will  be 


255.  To  find  the  longitude  by  a  lunar  distance  not  given  in  the 
Ephemeris.  —  The  regular  lunar-distance  stars  mentioned  in  Art. 
247  are  selected  nearly  in  the  moon's  path,  and  are  therefore  in 
general  most  favorable  for  the  accurate  determination  of  the 
Greenwich  time.  Nevertheless,  it  may  occasionally  be  found 
expedient  to  employ  other  stars,  not  too  far  from  the  ecliptic. 
Sometimes,  too,  a  different  star  may  have  been  observed  by 
mistake,  and  it  may  be  important  to  make  use  of  the  observation. 

VOL.  L—  27 


418  LONGITUDE. 

The  true  distance  d  is  to  be  found  from  the  observed  distance 
by  the  preceding  methods,  as  in  any  other  case.  Let  the  local 
time  of  the  observation  be  JJ  and  the  assumed  longitude  L. 
Take  from  the  Ephemeris  the  moon's  right  ascension  a  and  de- 
clination d  for  the  Greenwich  time  T  -f  L,  and  also  the  star's 
right  ascension  A  and  declination  A  ;  with  which  the  correspond- 
ing true  distance  d^  is  found  by  the  formula 

cos  d0  =  sin  d  sin  A  -f-  cos  d  cos  A  cos  (a  —  A) 

Then,  if  d  =  dQ,  the  assumed  longitude  is  correct  ;  if  otherwise, 
put 

A  —  the  increase  of  a  in  one  minute  of  mean  time, 
/?  =  the  increase  of  d     "  "  "        " 

•   f  =  the  increase  of  d     "  "  "         " 

then  we  have,  by  (464), 

cos  d  cos  A  sin  (a  —  A)  cos  d  sin  A  —  sin  3  cos  A  cos  (a  —  Jl) 

sin  </0  sin  dQ 

and  hence  the  correction  of  the  assumed  longitude  in  seconds 
of  time, 


For  computation  by  logarithms,  these  formulae  may  be  ar- 

ranged as  follows  : 

tan  A 
tanJf  = 


COS  (a  —  A) 

sin  A  cos  («5  —  M ) 

sin  M 

(466) 

cos  d  cos  A  sin  (a  —  A~) 

—!-  -f /J.eot  d0tan(d  —  Jf) 


r 

EXAMPLE. — Suppose  an  observer  has  measured  the  distance 
of  the  moon  from  Arcturus,  at  the  local  mean  time  1856  March 
16,  T=  10*  30*  0%  in  the  assumed  longitude  L  =  6h  Om  0s,  and, 
reducing  his  observation,  finds  the  true  distance 

d  =  73°  55'  10" 
what  is  the  true  longitude  ? 


BY    LUNAR    DISTANCES.  419 

For  the  Greenwich  time  T  -f  L  =  16*  30nl  we  find 

a  ==        8*  47-  G-.54  d  =  .+  23°  12'    7".l  A  =  -f  81".40 

A=      14     9    7.04          J  =  -f  19    55  44  .8  /?=-.    8.62 

a  —  A  =  —  5*  22*  0-.50  ==  —  80°  30'    7".5 

with  which  we  find  by  (466), 

dQ  =       73°  55'  35".  r  =  —  25".59 

«*_  do  =  _  25"  ^  =  +  58s  .6 

and  therefore  the  longitude  is  6*  0'*  58*.6. 

• 

256.  In  order  to  eliminate  as  far  as  possible  any  constant 
errors  of  the  instrument  used  in  measuring  the  distance,  we 
should  observe  distances  from  stars  both  east  and  west  of  the 
moon.  If  the  index  correction  of  the  sextant  is  in  error,  the 
errors  produced  in  the  computed  Greenwich  time,  and  conse- 
quently in  the  longitude,  will  have  different  signs  for  the  two 
observations,  and  will  be  very  nearly  equal  numerically:  they  will 
therefore  be  nearly  eliminated  in  the  mean.  If,  moreover,  the 
distances  are  nearly  equal,  the  eccentricity  of  the  sextant  will 
have  nearly  the  same  effect  upon  each  distance,  and  will  there- 
fore be  eliminated  at  the  same  time  with  the  index  error.  Since 
even  the  best  sextants  are  liable  to  an  error  of  eccentricity  of  as 
much  as  20",  according  to  the  confession  of  the  most  skilful 
makers,  and  this  error  is  not  readily  determined,  it  is  important 
to  eliminate  it  in  this  manner  whenever  practicable.  If  a  circle 
of  reflexion  is  employed  which  is  read  off  by  two  opposite 
verniers,  the  eccentricity  is  eliminated  from  each  observation ; 
but  even  with  suoh  an  instrument  the  same  method  of  observa- 
tion should  be  followed,  in  order  to  eliminate  other  constant 
errors. 

It  has  been  stated  by  some  writers  that  by  observing  distances 
of  stars  on  opposite  sides  of  the  moon  we  also  eliminate  a  con- 
stant error  of  observation,  such,  for  example,  as  arises  from  a 
faulty  habit  of  the  observer  in  making  the  contact  of  the  moon's 
limb  with  the  star.  This,  however,  is  a  mistake;  for  if  the 
habit  of  the  observer  is  to  make  the  contact  too  close,  that  is,  to 
bring  the  reflected  image  of  the  moon's  limb  somewhat  over 
the  star,  the  effect  will  be  to  increase  a  distance  on  one  side  of 
the  moon  while  it  diminishes  that  on  the  opposite  side,  and  the 
effect  upon  the  deduced  Greenwich  time  will  be  the  same  in 


420  LONGITUDE. 

both  cases      This  will  be  evident  from  the  following  diagram, 

(Fig.  31).     Suppose  a  and  b 
are    the    two    stars,    M   the 
moon's  limb.    If  the  observer 
*  /*  *  judges  a  contact  to  exist  when 

the  star  appears  within  the 
moon's  disc  as  at  c,  the  distance 
ac  is  too  small  and  the  distance 

be  too  great.  But,  supposing  the  moon  to  be  moving  in  the  direc- 
tion from  a  to  6,  each  distance  will  give  too  early  a  Greenwich 
time,  for  each  will  give  the  time  when  the  moon's  limb  was 
actually  at  c. 

If,  however,  we  observe  the  sun  in  both  positions,  this  kind 
of  error,  if  really  constant,  will  be  eliminated ;  for,  the  moon's 
bright  limb  being  always  turned  towards  the  sun,  the  error  will 
increase  both  distances,  and  will  produce  errors  of  opposite  sign 
in  the  Greenwich  time.  Hence,  if  a  series  of  lunar  distances 
from  the  sun  has  been  observed,  it  will  be  advisable  to  form  two 
distinct  means, — one,  of  all  the  results  obtained  from  increasing 
distances,  the  other,  of  all  those  obtained  from  decreasing  dis- 
tances :  the  mean  of  these  means  will  be  nearly  or  quite  free 
from  a  constant  error  of  observation,  and  also  from  constant  in- 
strumental errors. 

FINDING    THE    LONGITUDE   AT    SEA. 

257.  By  chronometers. — This  method  is  now  in  almost  universal 
use.  The  form  under  which  it  is  applied  at  sea  differs  very 
slightly  from  that  given  in  Art.  214.  The  correction  of  the 
chronometer  on  the  time  of  the  first  meridian  (that  of  Green- 
wich among  American  and  English  navigators)  is  found  at  any 
place  whose  longitude  is  known,  and  at  the  same  time  also  its 
daily  rate  is  to  be  established  with  all  possible  care.  The  rate 
being  duly  allowed  for  from  day  to  day  during  the  voyage,  the 
Greenwich  time  is  constantly  known,  and  therefore  at  any 
instant  when  the  local  time  is  obtained  by  observation,  the  lon- 
gitude of  the  ship  is  determined. 

The  local  time  on  shipboard  is  always  found  from  an  altitude 
of  some  celestial  object,  observed  with  the  sextant  from  the  sea 
horizon.  (Art.  156.)  The  computation  of  the  hour  angle  is 
then  made  by  (268),  and  the  resulting  local  time  is  compared 
directly  with  the  Greenwich  time  given  by  the  chronometer  at 


AT    SEA. 


421 


the  instant  of  the  observation.  The  data  from  the  Ephemeris 
required  in  computing  the  local  time  are  taken  for  the  Greenwich 
time  given  by  the  chronometer. 

EXAMPLE. — A  ship  being  about  to  sail  from  New  York,  the 
master  determined  the  correction  on  Greenwich  time  and  the 
rate  of  his  chronometer  by  observations  on  two  dates,  as  follows : 

1860  April  22,  at  Greenwich  noon,  chron.  correction  =  -f  3m  108.9 
«         "     30,  "  "         "  "  =  -f  3    43 .6 

Rate  in  8  days    =       -f  33  .6 
Daily  rate  -f-    4  .2 

On  May  18  following,  about  7A  807"  A.M.,  the  ship  being  in  lati- 
tude 41°  33'  N.,  three  altitudes  of  the  sun's  lower  limb  were 
observed  from  the  sea  horizon  as  below.  The  correction  of  the 
chronometer  on  that  day  is  found  from  the  correction  on  April  30 
by  adding  the  rate  for  18  days.  (It  will  not  usually  be  worth 
while  to  regard  the  fraction  of  a  day  in  computing  the  total  rate 
at  sea.)  The  record  of  the  observation  and  the  whole  computa- 
tion may  be  arranged  as  follows  : 


I860  May  18.  <j>  =  41 


Chronometer 

Mean 
Correction 

9*  37"  21'. 
"   37    53. 
"   38   20. 

=    9   37    51.3 
=  -f     4    59.2 

Gr.  date  =  May  17,  21  42  50.5 
for  which  time  we  take  from  the 
Ephemeris  the  quantities 

Q's  6  =  19°  38'  39" 
Semidiameter  =  15'  50" 
Equation  of  time  =  —  3m  49».8 


0  29°  40'  10" 
~  "    46    0 
"    50  50 

Mean  =  29  45  40 
Index  corr.  =  —  1  10 
Dip  =—42 


Barom.  30.32 '». 
Therm.  59°  F. 


k 

Semid. 
Refraction 
Parallax 
*                            h 

29    40  28      ^ 
=  +    15  50 
=  —      1  42 
=  +           8 

,      JUWM  W* 
^S^*^r  (TjtUf  jfi^ 

=  29    54  44 

t 

=  41    33     0 

sec  0.12588 

P 

s 

=  70    21  21 

cosec   0.02604 
cos  9.51464 

=  70    54  33 

s  —  h 

=  40    59  49 

sin  9.81692 

Apparent  time 
Eq.  of  time 
Local  mean  time 

=    7A32m    6'.  3 
=  -3    49.8 

9.48348 

sin   9.74174 

==  19  28    16.5 

Gr.          «       « 
Longitude 

=  21   42    50.5 

33°  38'.  5  W. 

=    2   14    34     = 

In  this  observation,  the  sun  was  near  the  prime  vertical,  a  posi- 
tion most  favorable  to  accuracy  (Art.  149). 


422  LONGITUDE. 

The  method  by  equal  altitudes  may  also  be  used  for  finding 
the  time  at  sea  in  low  latitudes,  as  in  Arts.  158,  159. 

258.  In  order  that  the  longitude  thus  found  shall  be  worthy 
of  confidence,  the  greatest  care  must  be   bestowed   upon  the 
determination    of  the   rate.     As   a   single    chronometer   might 
deviate  very  greatly  without  being  distrusted  by  the  navigator, 
it  is  well  to  have  at  least  three  chronometers,  and  to  take  the 
mean  of  the  longitudes  which  they  severally  give  in  every  case. 

But,  whatever  care  may  have  .been  taken  in  determining  the 
rate  on  shore,  the  sea  rate  will  generally  be  found  to  differ  from 
it  more  or  less,  as  the  instrument  is  affected  by  the  motion  of  the 
ship ;  and,  since  a  cause  which  accelerates  or  retards  one  chro- 
nometer may  produce  the  same  effect  upon  the  others,  the  agree- 
ment of  even  three  chronometers  is  not  an  absolutely  certain 
proof  of  their  correctness.  The  sea  rate  may  be  found  by 
determining  the  chronometer  correction  at  two  ports  whose 
difference  of  longitude  is  well  known,  although  the  absolute 
longitudes  of  both  ports  may  be  somewhat  uncertain.  For  this 
purpose,  a  "Table  of  Chronometric  Differences  of  Longitude"  is 
given  in  RAPER'S  Practice  of  Navigation,  the  use  of  which  is 
illustrated  in  the  following  example. 

EXAMPLE. — At  St.  Helena,  May  2,  the  correction  of  a  chro- 
nometer on  the  local  time  was  —  0*  23"1  10S.3.  At  the  Cape  of 
Good  Hope,  May  17,  the  correction  on  the  local  time  was 
-f  lh  14™  28*.6  ;  what  was  the  sea  rate  ? 

We  have 

Corr.  at  St.  Helena,  May  2d  =  —  0*  23™  10'.3 

Chron.  diff.  of  long,  from  Eaper      =4-1    36    45  . 
Corr.  for  Cape  of  G.  II.,  May  2d       =+  1    13    34.7 
«  «  «          "    17th  =  H-  1    14    28 .6 

Eate  in  15  days  =  -f    53  .9 

Daily  sea  rate     =  -{-      3  .59 

259.  By   lunar    distances. — Chronometers,    however    perfectly 
made,  are  liable  to  derangement,  and  cannot  be  implicitly  relied 
upon  in  a  long  voyage.     The  method  of  lunar  distances  (Arts. 
247-256)  is,  therefore,  employed  as  an  occasional  check  upon  the 
chronometers   even  where   the   latter  are  used  for  finding  the 
longitude  from  day  to  day.     When  there  is  no  chronometer  on 


AT   SEA.  423 

board,  the  method  of  lunar  distances  is  the  only  regularly  avail- 
able method  for  finding  the  longitude  at  sea,  at  once  sufficient!} 
accurate  and  sufficiently  simple. 

As  a  check  upon  the  chronometer,  the  result  of  a  lunar  distance 
is  used  as  in  Art.  253. 

In  long  voyages  an  assiduous  observer  may  determine  the  sea 
rates  of  his  chronometers  with  considerable  precision.  For  this 
purpose,  it  is  expedient  to  combine  observations  taken  at  various 
times  during  a  lunation  in  such  a  manner  as  to  eliminate  as  far 
as  possible  constant  errors  of  the  sextant  and  of  the  observer  (Art. 
256).  Suppose  distances  of  the  sun  are  employed  exclusively. 
Let  two  chronometer  corrections  be  found  from  two  nearly  equal 
distances  measured  on  opposite  sides  of  the  sun  on  two  different 
dates,  in  the  first  and  second  half  of  the  lunation  respectively. 
The  mean  of  these  corrections  will  be  the  correction  for  the 
mean  date,  very  nearly  free  from  constant  instrumental  and 
personal  errors.  In  like  manner,  any  number  of  pairs  of  equal, 
or  nearly  equal,  distances  may  be  combined,  and  a  mean  chro- 
nometer correction  determined  for  a  mean  date  from  all  the 
observations  of  the  lunation.  The  sea  rate  will  be  found  by 
comparing  twro  corrections  thus  determined  in  two  different 
lunations.  This  method  has  been  successfully  applied  in  voyages 
between  England  and  India. 

260.  By  the  eclipses  of  Jupiter's  satellites. — An  observed  eclipse 
of  one  of  Jupiter's  satellites  furnishes  immediately  the  Green- 
wich time  without  any  computation  (Art.  225.)     But  the  eclipse 
is  not  sufficiently  instantaneous  to  give  great  accuracy ;  for,  with 
the  ordinary  spy-glass  with  which  the  eclipse  may  be  observed 
on  board  ship,  the  time  of  the  disappearance  of  the  satellite  may 
precede  the  true  time  of  total  eclipse  by  even  a  whole  minute. 
The  time  of  disappearance  will  also  vary  with  the  clearness  of 
the  atmosphere.     Since,  however,  the  same  causes  which  accele- 
rate  the   disappearance  will   retard   the   reappearance,  if  both 
phenomena  are  observed  on  the  same  evening  under  nearly  the 
same   atmospheric    conditions,  the    mean  of  the  two  resulting 
longitudes  will  be  nearly  correct.     Still,  the  method  has  not  the 
advantage  possessed  by  lunar  distances  of  being  almost  always 
available  at  times  suited  to  the  convenience  of  the  navigator. 

261.  By  the  moori s  altitude. — This  method,  as  given  in  Art.  243,. 


424  CIRCLES    OF    POSITION. 

may  be  used  at  sea  in  low  latitudes ;  but,  on  account  of  the 
unavoidable  inaccuracy  of  an  altitude  observed  from  the  sea 
horizon,  it  is  even  less  accurate  than  the  method  of  the  preceding 
article,  and  always  far  inferior  to  the  method  of  lunar  distances, 
although  on  shore  it  is  one  which  admits  of  a  high  degree  of 
precision  when  carried  out  as  in  Art.  245. 

262.  By  occultations  of  stars  by  the  moon. — This  method,  which 
will  be  treated  of  in  the  chapter  on  eclipses,  may  be  successfully 
used  at  sea,  as  the  disappearance  of  a  star  behind  the  moon's 
limb  may  be  observed  with  a  common  spy-glass  at  sea  with 
nearly  as  great  a  degree  of  precision  as  on  shore  ;  but,  on  account 
of  the  length  of  the  preliminary  computations  as  well  as  of  the 
subsequent  reduction  of  the  observation,  it  is  seldom  that  a 
navigator  would  think  of  resorting  to  it  as  a  substitute  for  the 
convenient  method  of  lunar  distances. 


CHAPTER    VIII. 

FINDING   A   SHIP'S   PLACE   AT   SEA   BY   CIRCLES  OF   POSITION. 

263.  IN  the  preceding  two  chapters  we  have  treated  of 
methods  of  finding  the  position  of  a  point  on  the  earth's  surface 
by  tho.  two  co-ordinates  latitude  and  longitude  ;  and  therefore  in  all 
these  methods  the  required  position  is  determined  by  the  inter- 
section of  two  circles,  one  a  parallel  of  latitude  and  the  other  a 
meridian.  In  the  following  method  it  is  determined  by  circles 
oblique  to  the  parallels  of  latitude  and  the  meridians.  The  prin- 
ciple which  underlies  the  method  has  often  been  applied;  but  its 
value  as  a  practical  nautical  method  was  first  clearly  shown  by 
Capt.  THOMAS  H.  SUMNER.* 

Let  an  altitude  of  the  sun  (or  any  other  object)  be  observed 
at  any  time,  the  time  being  noted  by  a  chronometer  regulated  to 
Greenwich  time.  Suppose  that  at  this  Greenwich  time  the  sun 

*  A  new  and  accurate  method  of  finding  a  ship's  position  at  sea  by  projection  on  Merca- 
;tor's  chart:  by  Capt.  THOMAS  H.  SUMNER.  Boston,  1843. 


SUMNER'S  METHOD.  425 

is  vertical  to  an  observer  at  the  point  M  of  the  globe  (Fig  32). 
Let  a  small   circle  AA'A"  be  described  on 
the  globe  from  M  as  a  pole,  with  a  polar  dis-  Fis-  32- 

tance  MA  equal  to  the  zenith  distance,  or 
complement  of  the  observed  altitude,  of  the 
sun.  It  is  evident  that  at  all  places  within 
this  circle  an  observer  would  at  the  given 
time  observe  a  smaller  zenith  distance,  and 
at  all  places  without  this  circle  a  greater 
zenith  distance ;  and  therefore  the  observa- 
tion fully  determines  the  observer  to  be  on 
the  circumference  of  the  small  circle  AA' A" .  If,  then,  the 
navigator  can  project  this  small  circle  upon  an  artificial  globe  or 
a  chart,  the  knowledge  that  he  is  upon  this  circle  will  be  just  as  valuable 
to  him  in  enabling  him  to  avoid  dangers  as  the  knowledge  of  either  his 
latitude  alone  or  his  longitude  alone;  since  one  of  the  latter  elements 
only  determines  a  point  to  be  in  a  certain  circle,  without  fixing 
upon  any  particular  point  of  that  circle. 

The  small  circle  of  the  globe  described  from  the  projection  of 
the  celestial  object  as  a  pole  we  shall  call  a  circle  of  position. 

264.  To  find  the  place  on  the  globe  at  which  the  sun  is  vertical  (or  the 
sun's  projection  on  the  globe)  at  a  given  Greenwich  time. — The  sun's 
hour   angle   from   the    Greenwich   meridian  is  the    Greenwich 
apparent  time.     The  diurnal  motion  of  the  earth  brings  the  sun 
into  the  zenith  of  all  the  places  whose  latitude  is  just  equal  to 
the  sun's  declination.     Hence   the    required   projection  of  the 
sun  is  a  place  whose  longitude  (reckoned  westward  from  Green- 
wich from  0A  to  24*)  is  equal  to  the  Greenwich  apparent  time, 
and  whose  latitude  is  equal  to  the  sun's  declination  at  that  time. 

265.  From  an  altitude  of  the  sun  taken  at  a  given  Greenwich  time, 
to  find  the  circle  of  position  of  the  observer,  by  projection  on  an  artificial 
globe. — Find  the  Greenwich  apparent  time  and  the  sun's  declina- 
tion, and  put  down  on  the  globe  the  sun's   projection 'by  the 
preceding  article.     From  this  point  as  a  pole,  describe  a  small 
circle  with  a  circular  radius  equal  to  the  true  zenith  distance 
deduced  from  the  observation.     This  will  be  the  required  circle 
or  position. 

266.  The  preceding  problem  may  be  extended  to  any  celestial 


42^  CIRCLES   OF   POSITION. 

object.  The  pole  of  the  circle  of  position  will  always  be  the 
place  whose  west  longitude  is  the  Greenwich  hour  angle  of  the 
object  (reckoned  from  0*  to  24*)  and  whose  latitude  is  the  decli- 
nation of  the  object.  The  hour  angle  is  found  by  Art.  54. 

267.  To  find  both  the  latitude  and  the  longitude  of  a  ship  by  circles  of 
position  projected  on  an  artificial  globe. — First.  Take  the  altitudes 
of  two    different   objects  at  the    same  time  by  the  Greenwich 
chronometer.    Put  down  on  the  globe,  by  the  preceding  problem, 
their  two  circles  of  position.     The  observer,  being  in  the  circum- 
ference of  each  of  these  circles,  must  be  at  one  of  their  two  points 
of  intersection ;  which  of  the  two,  he  can  generally  determine 
from  an  approximate  knowledge  of  his  position. 

Second.  Let  the  same  object  be  observed  at  two  different  times, 
and  project  a  circle  of  position  for  each.  Their  intersection 
gives  the  position  of  the  ship  as  before.  If  between  the  observa- 
tions the  ship  has  moved,  the  first  altitude  must  be  reduced  to 
the  second  place  of  observation  by  applying  the  correction  of 
Art.  209,  formula  (880).  The  projection  then  gives  the  ship's 
position  at  the  second  observation. 

268.  From  an  altitude  of  a  celestial  body  taken  at  a  given  Greenwich 
time,  to  find  the  circle  of  position  of  the  observer,  by  projection  on  a 
Mercator  chart. — The  scale  upon  which  the  largest  artificial  globes 
are  constructed  is  much  smaller  than  that  of  the  working  charts 
used  by  navigators.     But  on   the  Mercator   chart   a   circle  of 

position  will  be  distorted,  and  can  only 
Fig.  33.  be  laid  down  by  points.     Let  L,  L',  Ln 

(Fig.  33)  be  any  parallels  of  latitude 
_  L,f  crossed  by  the  required  circle.  For  each 
of  these  latitudes,  with  the  true  altitude 
found  from  the  observation  and  the  polar 
distance  of  the  celestial  body  taken  for 
the  Greenwich  time,  compute  the  local 
time,  and  hence  the  longitude,  "  by  chro- 
nometer" (Art.  257).  Let  I,  I',  I"  be  the 
longitudes  thus  found.  Let  A,  A',  A"  be  the  points  whose 
latitudes  and  longitudes  are,  respectively,  L,  1;  L' ,  V ;  L" ,  I"  ; 
these  are  evidently  points  of  the  required  circle.  The  ship  is 
consequently  in  the  curve  AA'A",  traced  through  these 
points. 


I    V      I" 


SUMNER'S  METHOD.  427 

In  practice  it  is  generally  sufficient  to  lay  down  only  two 
points ;  for,  the  approximate  position  of  the  ship  being  known, 
if  L  and  Lf  are  two  latitudes  between  which  the  ship  may  be 
assumed  to  be,  her  position  is  known  to  be  on  the  curve  AA' 
somewhere  between  A  and  A'.  When  the  difference  between 
L  and  Lr  is  small,  the  arc  AA'  will  appear  on  the  chart  as  a 
straight  line. 

269.  To  jind  the  latitude  and  longitude  of  a  ship  by  circles  of  position 
projected  on  a  Mcrcator  chart. — First.  Let  the  altitudes  of  two 
objects  be  taken  at  the  same  time.  Assume  two  latitudes  em- 
bracing between  them  the  ship's  probable  position,  and  find  two 
points  of  each  of  their  two  circles  of  position  by  the  preceding 
problem,  and  project  these  points  on  the  chart.  Each  pair  of 
points  being  joined  by  a  straight  line, 
the  intersection  of  the  two  lines  is  B>  '  A> 

very  nearly  the  ship's  position.     Thus, 
if  one  object  gives  the   points  A,  Af 

(Fig.    34)    corresponding   to    the    lati-    

tudes  .L,  Lf,  and  the  other  object  the 

points  J9,  E'  corresponding  to   the  same   latitudes,  the   ship's 

position  is  the  point  (7,  the  intersection  of  AA'  and  BB'. 

It  is,  of  course,  not  essential  that  the  same  latitudes  should  be 
used  in  computing  the  points  of  the  two  circles ;  but  it  is  more 
convenient,  and  saves  some  logarithms. 

If  greater  accuracy  is  desired,  the  circles  may  be  more  fully 
laid  down  by  three  or  more  points  of  each. 

Second. — The  altitude  of  the  same  object  may  be  taken  at  two 
different  times,  and  the  circles  laid  down  as  before ;  the  usual 
reduction  of  the  first  altitude  being  applied  when  the  ship  changes 
her  position  between  the  observations. 

It  is  evident  from  the  nature  of  the  above  projection  that  the 
most  favorable  case  for  the  accurate  determination  of  the  inter- 
section C  is  that  in  which  the  circles  of  position  intersect  at  right 
angles.  Hence  the  two  objects  observed,  or  the  two  positions 
of  the  same  object,  should,  if  possible,  differ  about  90°  in  azimuth. 
This  agrees  with  the  results  of  the  analytical  discussion  of  the 
method  of  finding  the  latitude  by  two  altitudes,  Art.  183. 

If  the  chronometer  does  not  give  the  true  Greenwich  time,  the 
only  effect  of  the  error  will  be  to  shift  the  point  C  towards  the 
east  or  the  west,  without  changing  its  latitude,  unless  the  error  is 


428  CIRCLES    OF    POSITION. 

so  great  as  to  affect  sensibly  the  declination  which  is  taken  from 
the  Ephemeris  for  the  time  given  by  the  chronometer.  This  method 
is,  therefore,  a  convenient  substitute  for  the  usual  method  of  find- 
ing the  latitude  at  sea  by  two  altitudes,  a  projection  on  the  sailing 
chart  being  always  sufficient  for  the  purposes  of  the  navigator. 

Instead  of  reducing  the  first  altitude  for  the  change  of  the  ship's 

position  between  the  observations,  we  may  put  down  the  circle 

of  position  for  each  observation  and  afterwards  shift  one  of  them 

F.  by  a  quantity  due  to  the  ship's  run. 

ff A'  a' t    Thus,  let  the  first  observation  give  the 

position  line  A  A'  (Fig.  35),  and  let  Aa 
represent,  in  direction  and  length,  the 
L     ship's  course  and   distance   sailed  be- 
tween   the    observations.      Draw    aa' 

parallel  to  AA'.  Then,  BE'  being  the  position  line  by  the 
second  observation,  its  intersection  C  with  aa'  is  the  required 
position  of  the  ship  at  the  second  observation. 

270.  If  the  latitude  is  desired  by  computation,  independently 
of  the  projection,  it  is  readily  found  as  follows.     Let 

Zj,  Z2  =  the  longitudes  (of  A  and  B)  found  from  the  first  and 
second  altitudes  respectively  with  the  latitude  L, 

Z^,  12'  =  the  longitudes  (of  A'  and  B'}  found  from  the  same 

altitudes  with  the  latitude  L'} 
L0  =  the  latitude  of  C. 

From  Fig.  34  we  have,  by  the  similarity  of  the  triangles  ABC 
and  J/jB'<7, 

l> £'.  ^ I  =  B' C  •  BC 

whence 

(  T,' /"A  (i n 

*L_Ji  (467) 


cy—yj +$—'*.) 

This  formula  reduces  SUMNER'S  method  of  "  double  altitudes" 
to  that  given  long  ago  by  LALANDE  (Astronomic,  Art.  399*2,  and 
Abrege  de  Navigation,  p.  68).  The  distinctive  feature  of  SUMNER'S 
process,  however,  is  that  a  single  altitude  taken  at  any  time  is 
made  available  for  determining  a  line  of  the  globe  on  which  the 
ship  is  situated. 


MERIDIAN    LINE.  429 

271.   To  find  the  azimuth  of  the  sun  by  a  position  line  projected  on 
the  chart. — Let  AA'  (Fig.  36)  be  a  position  line  on 
the  chart,  derived  from  an  observed  altitude  by 
Art.  268.     At  any  point  C  of  this  line  draw  CM 
perpendicular  to  A  A',  and  let  NCS  be  the  meri- 
dian passing  through  (7;  then  SCM  is  evidently 
the  sun's  azimuth.     The  line  CM  is,  of  course, 
drawn  on  that  side  of  the  meridian  NS  upon 
which  the  sun  was  known  to  be  at  the  time  of 
the  observation. 

The  solution  is  but  approximate,  since  AAf  should  be  a  curve 
line,  and  the  azimuth  of  the  normal  CM  would  be  different  for 
different  points  of  A  A'.  It  is,  however,  quite  accurate  enough 
for  the  purpose  of  determining  the  variation  of  the  compass  at 
sea,  which  is  the  only  practical  application  of  this  problem. 


CHAPTER   IX. 

THE   MERIDIAN   LINE   AND   VARIATION    OF   THE   COMPASS. 

272.  THE  meridian  line  is  the  intersection  of  the  plane  of  the 
meridian  with  the  plane  of  the  horizon.     Some  of  the  most  use- 
ful methods  of  finding  the  direction  of  this  line  will  here  be 
briefly  treated  of;  but  the  full  discussion  of  the  subject  belongs 
to  geodesy. 

273.  By  the  meridian  passage  of  a  star. — If  the  precise  instant 
when  a  star  arrives  at  its  greatest  altitude  could  be  accurately 
distinguished,  the  direction  of  the  star  at  that  instant,  referred 
to  the  horizon,  would  give  the  direction  of  the  meridian  line  ;  but 
the  altitude  varies  so  slowly  near  the  meridian  that  this  method 
only  serves  to  give  a  first  approximation. 

274.  By  shadows. — A  good  approximation  may  be  made  as 
follows.    Plant  a  stake  upon  a  level  piece  of  ground,  and  give  it 
a  vertical  position  by  means  of  a  plumb  line.     Describe  one  or 


430  MERIDIAN    LINE. 

more  concentric  circles  on  the  ground  from  the  foot  of  the  stake 
as  a  centre.  At  the  two  instants  before  and  after  noon  when  the 
shadow  of  the  stake  extends  to  the  same  circle,  the  azimuths  of 
the  shadow  east  and  west  are  equal.  The  points  of  the  circle  at 
which  the  shadow  terminates  at  these  instants  being  marked,  let 
the  included  arc  be  bisected  ;  the  point  of  bisection  and  the  centre 
of  the  stake  then  determine  the  meridian  line.  Theoretically,  a 
small  correction  should  be  made  for  the  sun's  change  of  declina- 
tion, but  it  would  be  quite  superfluous  in  this  method. 

275.  By  single  altitudes.  —  "With  an  altitude  and  azimuth  instru- 
ment, observe  the  altitude  of  a  star  at  the  instant  of  its  passage 
over  the  middle  vertical  thread  (at  any  time),  and  read  the 
horizontal  circle.  Correct  the  observed  altitude  for  refraction. 
Then,  if 

h  =  the  true  altitude, 

<p  =  the  latitude  of  the  place  of  observation, 

p  =  the  star's  polar  distance, 

A  =  the  star's  azimuth, 

A'  =  the  reading  of  the  horizontal  circle, 

we  have,  from  the  triangle  formed  by  the  zenith,  the  pole,  and 
the  star, 

tan*  j  A  =  Bin  («-?)..!..(«  -A) 

cos  s  cos  (s  —  p) 
in  which 


In  this  formula  the  latitude  may  be  taken  with  the  positive  sign, 
whether  north  or  south,  and  p  is  then  to  be  reckoned  from  the 
elevated  pole  ;  consequently,  also,  A  will  be  the  azimulh  reckoned 
from  the  elevated  pole. 

It  is  evident  that  in  order  to  bring  the  telescope  into  the  plane 
of  the  meridian  we  have  only  to  revolve  the  instrument  through 
the  angle  A,  and  therefore  either  A'  +  A  or  A'  —  A,  according 
to  the  direction  of  the  graduations  of  the  circle,  will  be  the 
reading  of  the  horizontal  circle  when  the  telescope  is  in  the 
meridian. 

The  same  method  can  be  followed  when  the  azimuth  is  ob- 
served with  a  compass  and  the  altitude  is  measured  with  a  sex- 
tant; and  then  A'  —  A  is  the  variation  of  the  compass. 


MERIDIAN    LINE.  431 

276.  From  the  first  equation  of  (50),  <p  and  d  being  constant, 
we  have 

HA  dk 

a  A  =  -- 

cos  h  tan  q 

and  therefore  an  error  in  the  observed  altitude  will  have  the 
least  effect  upon  the  computed  azimuth  when  tan  q  is  a  maxi- 
mum ;  that  is,  when  the  star  is  on  the  prime  vertical.  There- 
fore, in  the  practice  of  the  preceding  method  the  star  should  be 
as  far  from  the  meridian  as  possible. 

277.  By  equal  altitudes  of  a  star.  —  Observe  the  azimuth  of  a  star 
with  an  altitude  and  azimuth  instrument,  or  a  compass,  when  at 
the  same  altitude  east  and  west  of  the  meridian.     The  mean  of 
the  two  readings  of  the  instrument   is   th<3   reading  when  its 
sight   line    is   in   the    direction  of  the   meridian.     This  is  the 
method  of  Article  274,  rendered  accurate  by  the  introduction 
of  proper  instruments  for  observing  both  the  altitude  and  the 
azimuth. 

278.  If  equal  altitudes  of  the  sun,  are  employed,  a  correction 
for  the  change  of  the  sun's  declination  is  necessary,  since  equal 
azimuths  will  no  longer  correspond  to  equal  altitudes.     Let 

A'  =  the  east  azimuth  at  the  first  observation, 
A  =    "    west       «  "      •  second       " 

d  =  the  declination  at  noon, 

A'S  =  the  increase  of  declination  from  the  first  to  the  second 
observation, 

then,  by  (1),  we  have,  h  being  the  altitude  in  each  case, 

sin  (<5  —  $  A<5)  =  sin  <p  sin  h  —  cos  <p  cos  h  cos  A! 
sin  (5  -j-  \  Ad)  =  sin  <p  sin  h  —  cos  <p  cos  h  cos  A 

the  difference  of  which  gives 

2  cos  8  sin  £  A<S  =  2  cos  y  cos  h  sin  £  (  A  -f-  A')  sin  £  (  A  —  A'*) 

whence,  since  A<?  is  but  a  few  minutes,  we  have,  with  sufficient 
accuracy, 

A*  COS  * 


cos  <p  cos  h  sin  A 


432  MERIDIAN    LINE. 

It  will  be  necessary  to  note  the  times  of  the  two  observations 
in  order  to  find  &3.  If  we  take  half  the  elapsed  time  as  the 
hour  angle  t  of  the  western  observation,  we  shall  have,  instead 
of  (469),  the  more  convenient  formula 

A.-A'  =  -  ^—  (470) 

cos  <f>  sin  t 

It  will  not  be  necessary  to  know  the  exact  value  of  A,  if  only 
the  same  instrumental  altitude  is  employed  at  both  observations. 

Now  let  AI  and  Al  be  the  readings  of  the  horizontal  circle  at 
the  two  observations,  then  the  readings  corresponding  to  equal 

azimuths  are 

AJ  and  Al  —  (A  —  A') 

and,  consequently,  the  reading  for  the  meridian  is  the  mean  of 
these,  or 


That  is,  the  reading  for  the  meridian  is  the  mean  of  the  oh- 
served  readings  diminished  by  one-half  the  correction  (470). 
WQ  here  suppose  the  graduations  to  proceed  from  0°  to  360°, 
and  from  left  to  right. 

279.  By  the  angular  distance  of  the  sun  from  any  terrestrial  object.  — 
If  the  true  azimuth  of  any  object  in  view  is  known,  the  direction 
of  the  meridian  is,  of  course,  known  also.  The  following  method 
can  be  carried  out  with  the  sextant  alone.  Measure  the  angular 
distance  of  the  sun's  limb  from  any  well-defined  point  of  a 
distant  terrestrial  object,  and  note  the  time  by  a  chronometer. 
Measure  also  the  angular  height  of  the  terrestrial  point  above 
the  horizontal  plane.  The  correction  of  the  chronometer  being 
known,  deduce  the  local  apparent  time,  or  the  sun's  hour  angle  t 
(Art.  54),  and  then  with  the  sun's  declination  d  and  the  latitude  <p 
compute  the  true  altitude  h  and  azimuth  A  of  the  sun  by  the 
formulae  (16),  or 

tan  3  tan  t  cos  M  ,^N         A  ,A~-t^ 

tan  M  —  --  ,    tan  A  —  —  -  ,    tan  h  =  cot  (if  —  M)  cos  A  (471) 

cost  sin 


Now,  let  0,  Fig.  37,  be  the  apparent  position  of  the  terrestrial 
point,  projected  upon  the  celestial  sphere;  S  the  apparent  place 
of  the  sun,  Z  the  zenith,  P  the  pole  ;  and  put 


MERIDIAN    LINE. 


433 


D  =  the  apparent  angular  distance  of  the  FJg- 

sun's  centre  from  the  terrestrial  point 
=  the  observed  distance   increased  by 

the  sun's  semidiameter, 
H  =  the  apparent  altitude  of  the  point, 
h'  =  the  sun's  apparent  altitude, 
a  =  the  difference  of  the  azimuth  of  the 

sun  and  the  point, 
A'  =  the  azimuth  of  the  point. 

The  apparent  altitude  Tif  will  be  deduced  from  the  true  altitude* 
by  adding  the  refraction  and  subtracting  the  parallax.  Then  in 
the  triangle  SZO  we  have  given  the  three  sides  ZS  =  90°  —  A', 
ZO  =  90°  —  H,  SO  =  D,  and  hence  the  angle  SZO  =  a  can  be 

found  by  the  formula 

sin  (s  —  H}  sin  (s  —  h') 


tan2  £  a  = 


in  which 
Then  we  have 


cos  s  cos  (s 


(472) 


(473) 


and  the  proper  sign  of  a  to  be  used  in  this  equation  must  be 
determined  by  the  position  of  the  sun  with  respect  to  the  object 
at  the  time  of  the  observation. 

If  the  altitude  of  the  sun  is  observed,  we  can  dispense  with 
the  computation  of  (471),  and  compute  A  by  the  formula  (468). 
The  chronometer  will  not  then  be  required,  but  an  approximate 
knowledge  of  the  local  time  and  the  longitude  is  necessary  in 
order  to  find  3  from  the  Ephemeris. 

If  the  terrestrial  object  is  very  remote,  it  will  often  suffice  to 
regard  its  altitude  as  zero,  and  then  we  shall  find  that  (472) 

reduces  to 

tan  J  a  =  -/[tan  J  (D  -f  h')  tan  \(D  —  A')]  (474) 

This  method  is  frequently  used  in  hydrographic  surveying  to 
determine  the  meridian  line  of  the  chart. 

EXAMPLE. — From  a  certain  point  B  in  a  survey  the  azimuth 
of  a  point  C  is  required  from  the  following  observation : 


Chronometer  time       =-  4*  12m  12' 
Chronom.  correction  =  —   2      0 
Local  mean  time         =  4    10    12 
Equation  of  time         = —  4    10.9 
Local  app.  time,      t  =  4     6      1 .1 
VOL.   I.— 28 


Altitude  of  C  =  11=  0°  30'  20" 
Distance  of  the  nearest  limb  of  the 
sun  from  the  point  C  =  48°  17'  10" 
Semidiameter  =  16  1 

D  —  48    33  11 


434  MERIDIAN    LINE. 

The  sun's  declination  was  3  =  -f-  4°  16'  55",  the  latitude  was 
y  —  -f  38°  58'  50" ;  and  hence,  by  (471),  we  find 

A  =  74°  36'  36"  h  =  24°  37'  58" 

Refraction  and  parallax  =  1  54 

hf  =  24    39  52 
and,  by  (472), 

a  =  43°  35'  6" 

"Now,  the  sun  was  on  the  right  of  the  object,  and  hence 

A'=  A  — a  =  31°  1'30" 

Therefore,  a  line  drawn  on  the  chart  from  B  on  the  left  of  the 
line  BC,  making  with  it  the  angle  31°  V  30",  will  represent  the 
meridian. 

280.  By  two  measures  of  the  distance  of  the  sun  from  a  terrestrial 
object. — In  the  practice  of  the  preceding  method  with  the  sextant, 
it  is  not  always  practicable  to  measure  the  apparent  altitude  of 
the  terrestrial  object.     We  may  then  measure  the  distance  of 
the  sun  from  the  object  at  two  different  times,  and,  first  com- 
puting the  altitude  and  azimuth  of  the  sun  at  each  observation, 
we  may  from  these  data  compute  the  altitude  of  the  object  and 
the  difference  between  its  azimuth  and  that  of  the  sun  at  either 
observation,  by  formulae  entirely  analogous  to  those  employed 
in  computing  the  latitude  and  time  from  two  altitudes,  Art.  178, 
(304),  (305),  (306),  and  (307). 

281.  By  the  azimuth  of  a  star  at  a  given  time. — "When  the  time  is 
known,  the  azimuth  of  the  star  is  found  by  (471) :   hence  we 
have  only  to  direct  the  telescope  of  an  altitude  and  azimuth 
instrument  to  the  star  at  any  time,  and  then  compare  the  read- 
ing of  its  horizontal  circle  with  the  computed  azimutE. 

This  method  will  be  very  accurate  if  a  star  near  the  pole  is 
employed,  since  in  that  case  an  error  in  the  time  will  produce  a 
comparatively  small  error  in  the  azimuth.  It  will  be  most  accu- 
rate if  the  star  is  observed  at  its  greatest  elongation,  as  in  the 
following  article. 

282.  By  the  greatest  elongation   of  a  circumpolar  star. — At  the 
instant  of  the  greatest  elongation  we  have,  by  Art.  18, 

cos  d 

sin  A  =  - 


MERIDIAN    LINE.  435 

in  which  A  is  the  azimuth  reckoned  from  the  elevated  pole.  At 
this  instant  the  star's  azimuth  reaches  its  maximum,  and  for  & 
certain  small  interval  of  time  appears  to  be  stationary,  so  that 
the  observer  has  time  to  set  his  instrument  accurately  upon  the 
star. 

In  order  to  be  prepared  for  the  observation,  the  time  of  the 
elongation  must  be  (at  least  approximately)  known.  The  hour 
angle  of  the  star  is  found  by  the  formula 

tan  < 

cos  t  = 


tan  d 

and  from  t  and  the  star's  right  ascension  the  local  time  is  found, 
Art.  55. 

The  pole  star  is  preferred,  on  account  of  its  extremely  slow 
motion. 

If  the  latitude  is  unknown,  the  direction  of  the  meridian  may 
nevertheless  be  obtained  by  observing  the  star  at  both  its  eastern 
and  its  western  greatest  elongations.  The  mean  of  the  readings 
of  the  horizontal  circle  at  the  two  observations  is  the  reading  for 
the  meridian. 

283.  One  of  the  most  refined  methods  of  determining  the 
direction  of  the  meridian  is  that  by  which  the  transit  instrument 
is   adjusted,  or  by  which  its  deviation   from  the  plane  of  the 
meridian  is  measured  ;  for  which  see  Vol.  II. 

284.  At  sea,  the  direction  of  the  meridian,  or  the  variation  of 
the  compass,  is  found  with  sufficient  accuracy  by  the  graphic 
process  of  Art.  271. 


436  SOLAR   ECLIPSES. 


CHAPTER   X. 

ECLIPSES. 

285.  THE  term  eclipse,  in  astronomy,  may  be  applied  to  any 
obscuration,  total  or  partial,  of  the  light  of  one  celestial  body  by 
another.     But  the  term  solar  eclipse  is  usually  confined  to  an 
eclipse  of  the  sun  by  the  moon;  while  an  eclipse  of  the  sun  by 
one  of  the  inferior  planets  is  called  a  transit  of  the  planet.     An 
eclipse  of  a  star  or  a  planet  by  the  moon  is  called  an  occultation 
of  the  star  or  planet.     A  lunar  eclipse  is  an  eclipse  of  the  moon 
by  the  earth. 

All  these  phenomena  may  be  computed  upon  the  same  general 
principles  ;  and  the  investigation  of  solar  eclipses,  with  which  we 
shall  set  out,  will  involve  nearly  every  thing  required  in  the 
other  cases. 

SOLAR  ECLIPSES, 
PREDICTION   OF   SOLAR   ECLIPSES   FOR   THE   EARTH   GENERALLY. 

286.  For  the  purposes  of  general  prediction,  and  before  enter- 
ing upon  any  precise  computation,  it  is  convenient  to  know  the 
limits  which  determine  the  possibility  of  the  occurrence  of  an 
eclipse  for  any  part  of  the  earth.     These  limits  are  determined 
in  the  following  problem. 

287.  To  find  whether  near  a  given  conjunction  of  the  sun  and  moon, 
an  eclipse  of  the  sun  will  occur. — In  order  that  an  eclipse  may  occur, 

Fi    3g  the  moon  must  be  near  the  ecliptic,  and, 

therefore,   near  one  of  the  nodes  of  her 
or^it.     Let  NS  (Fig.  38)  be  the  ecliptic,  N 
the  moon's  node,  NM  the  moon's  orbit,  $ 
_      and  M  tne  centres  of  the  sun  and  moon  at 
the  time  of  conjunction   in   longitude,  so 
that  MS  is  a  part  of  a  circle  of  latitude  and  is  perpendicular  to 


GENERAL    PREDICTIONS.  437 

NS.    Let  Sf,  Mf,  be  the  centres  of  the  sun  and  moon  when  at 
their  least  true  distance,  and  put 

/?  =  the  moon's  latitude  at  conjunction  =  SM, 

1=  the  inclination  of  the  moon's  orbit  to  the  ecliptic, 

A  =  the  quotient  of  the  moon's  motion  in  longitude  divided 

by  the  sun's,     $'  F 

2—  the  least  true  distance  =  S'Mf, 
Y  =  the  angle  SMS'. 

We  may  regard  NMS  as  a  plane  triangle  ;  and,  drawing  MfP 
perpendicular  to  NS,  we  find 


SS'  =  ptany  SP=  A/9  tan  y 

and  hence 


S'P=P(1  —  1)  tanr  M'P=  p  — 

212  =  p  [(A  —  I)2  tan2  r  +  (1  —  A  tan  I  tan  r)2] 

To  find  the  value  of  y  for  which  this  expression  becomes  a  mini- 
mum, we  put  its  derivative  taken  relatively  to  y  equal  to  zero, 
whence 

/I  tan/ 

tan  y  =  - 
(A  —  I)2  -[-  A2  tan2  1 

which  substituted  in  the  value  of  I2  reduces  it  to 


(A  —  l)2-f  A2  tan2  1 
If  then  we  assume  I'  such  that 

tan  I1  =  -      -  tan  J  (475) 

we  have  for  the  least  true  distance 

J  =  /9  cos  J'  (476) 

The  apparent  distance  of  the  centres  of  the  sun  and  moon  as 
seen  from  the  surface  of  the  earth  may  be  less  than  I  by  the 
difference  of  the  horizontal  parallaxes  of  the  two  bodies  :  so  that 

if  we  put 

l 

TT  =  the  moon's  horizontal  parallax, 
n'=  the  sun's  "  " 


438 
we  have 


SOLAR    ECLIPSES. 


minimum  apparent  distance  =  2  —  (?r  — wf 


An  eclipse  will  occur  when  this  least  apparent  distance  of  the 
centres  is  less  than  the  sum  of  the  semi  diameters  of  the  bodies; 
and  therefore,  putting 

$  =  the  moon's  semidiameter, 
y=  the  sun's  " 

we  shall  have,  in  case  of  eclipse, 

2  —  (5  —  ;:')<  s  -f  Sr 


or 


/?CQSI'<  rr  —  r'-f  S  -f 


(477) 


This  formula  gives  the  .required  limit  with  great  precision ; 
hut,  since  1'  is  small,  its  cosirie  does  not  vary  much  for  different 
eclipses,  and  we  may  in  most  cases  employ  its  mean  value.  We 
have,  hy  observation, 


Greatest  values. 

Least  values. 

Mean  values. 

I 

5°  20'     6" 

4°  57'  22" 

50     3,44,, 

7t 

61'  32" 

52'  50" 

57'  11" 

TT' 

9 

8 

8.5 

S 

16  46 

14  24 

15  35 

s' 

16  18 

15  45 

16     1 

/I 

16.19 

10.89 

13.5 

From  the  mean  values  of  I  and  ^  we  find  the  mean  value  of 
sec  I'  =  1.00472,  and  the  condition  (477)  becomes 


/5  <  (-  —  ;:'  -f-  s  +  s')  X  1.00472 


or 


ft  <  * 


S  -f  Sf  +  (rr  — 


«')  X  -00472 


where  the  small  fractional  term  varies  between  20"  and  30". 
Taking  its  mean  value,  we  have,  with  sufficient  precision  for  all 
but  very  unusual  cases, 


^V  -f  *  -f  V  -f  25" 


(478) 


FUNDAMENTAL   EQUATIONS.  439 

If  in  this  formula  we  substitute  the  greatest  values  of  sr,  s, 
and  s',  and  the  least  value  of  TT',  the  limit 

/?  <  1°  34'  53"  /  *  3  " 

is  the  greatest  limit  of  the  moon's  latitude  at  the  time  of  con- 
junction, for  which  an  eclipse  can  occur. 

If  in  (478)  we  substitute  the  least  values  of  TT,  s,  and  s',  and 
the  greatest  value  of  it',  the  limit 

0  <  1°  23'  15" 

is  the  least  limit  of  the  moon's  latitude  at  the  time  of  conjunc- 
tion for  which  an  eclipse  can  fail  to  occur.          (jJiX 

Hence  a  solar  eclipse  is  certain  if  at  new  moon  /9<  1°  23'  15", 
impossible  if  /3>  1°  34'  53",  and  doubtful  between  these  limits.  For 
the  doubtful  cases  we  must  apply  (478),  or  for  greater  precision 
(477),  using  the  actual  values  of  TT,  TT',  s,  s',  ^,  and  I  for  the  date. 

EXAMPLE. — On  July  18,  1860,  the  conjunction  of  the  moon 
and  sun  in  longitude  occurs  at  2A  19"\2  Greenwich  mean  time: 
will  an  eclipse  occur?  We  find  at  this  time,  from  the  Ephemeris, 

0  =  0°  33'  18".6 

which,  being  within  the  limit  1°  23'  15",  renders  an  eclipse  cer- 
tain at  this  time. 

Having  thus  found  that  an  eclipse  will  be  visible  in  some  part 
of  the  earth,  we  can  proceed  to  the  exact  computation  of  the 
phenomenon.  The  method  here  adopted  is  a  modified  form  of 
BESSEL'S,*  which  is  at  once  rigorous  in  theory  and  simple  in 
practice.  For  the  sake  of  clearness,  I  shall  develop  it  in  a  series 
of  problems. 

Fundamental  Equations  of  the  Theory  of  Eclipses. 

288.  To  investigate  the  condition  of  the  beginning  or  ending  of  a  solar 
eclipse  at  a  given  place  on  the  earth's  surface. — The  observer  sees  the 
limbs  of  the  sun  and  moon  in  apparent  contact  when  he  is  situated 
in  the  surface  of  a  cone  which  envelops  and  is  in  contact  with 
the  two  bodies.  We  may  have  two  such  cones : 

*  See  Astronomische  Nachrichten,  Nos.  151,  152,  and,  for  the  full  development  of  the 
method  with  the  utmost  rigor,  BESSEL'S  Astronomische  Untersuchungen,  Vol.  II. 
HANSEN'S  development,  based  upon  the  same  fundamental  equations,  but  theoreti- 
cally less  accurate,  may  also  be  consulted  with  advantage:  it  is  given  in  Astronom. 
Nach.,  Nos.  339-342. 


440 


SOLAR    ECLIPSES. 


First.  The  cone  whose  vertex  falls  between  the  sun  and  the 
moon,  as  at  V9  Fig.  39,  and  which  is  called  the  peimmbral  cone. 
An  observer  at  C,  in  one  of  the  elements  CB  V  of  the  cone,  sees 
the  points  A  and  B  of  the  limbs  of  the  sun  and  moon  in  apparent 
exterior  contact,  which  is  either  the  first  or  the  last  contact;  that 
is,  either  the  beginning  or  the  ending  of  the  whole  eclipse. 


Fig.  39. 


Fig.  40. 


E  F 


\ 


Second.  The  cone  whose  vertex  is  beyond  the  moon  (in  the 
direction  of  the  earth),  as  at  V,  Fig.  40,  and  which  is  called  the 
wnbral  cone,  or  cone  of  total  shadow.  An  observer  at  (7,  in  the 
element  CVBA,  sees  the  points  A  and  B  of  the  limbs  of  the  sun 
and  moon  in  apparent  interior  contact,  which  is  the  beginning  or 
the  ending  of  annular  eclipse  in  case  the  observer  is  farther 
from  the  moon  than  the  vertex  of  the  cone  (as  in  the  figure),  and 
which  is  either  the  beginning  or  the  ending  of  total  eclipse  in 
case  the  observer  is  between  the  vertex  of  the  cone  and  the 
moon. 

If  now  a  plane  is  passed  through  the  point  C9  at  right  angles 
to  the  axis '  SVD  of  the  cone,  its  intersection  with  the  cone  will 


FUNDAMENTAL    EQUATIONS.  441 

be  a  circle  (the  sun  and  moon  being  regarded  as  spherical)  whose 
radius,  (7Z),  we  shall  call  the  radius  of  the  shadow  (penumbral  or 
umbral)  for  that  point.  The  condition  of  the  occurrence  of  one 
of  the  above  phases  to  an  observer  is,  then,  that  the  distance  of 
the  point  of  observation  from  the  axis  of  the  shadow  is  equal  to  the 
radius  of  the  shadow  for  that  point.  The  problems  which  follow 
will  enable  us  to  translate  this  condition  into  analytical  language. 

289.  To  find  for  any  given  time  the  position  of  the  axis  of  the 
shadow. — The  axis  of  the  cone  of  shadow  produced  to  the  celes- 
tial sphere  meets  it  in  that  point  in  which  the  sun  would  be 
projected  upon  the  sphere  by  an  observer  at  the  centre  of  the 
moon.  Let  (9,  Fig.  41,  be  the  centre  of 
the  earth ;  $,  that  of  the  sun  ;  M,  that  of 
the  moon.  The  line  MS  produced  to 
the  infinite  celestial  sphere  meets  it  in 
the  common  vanishing  point  of  all  lines 
parallel  to  MS;  that  is,  in  the  point  Z,  in 
which  the  line  OZ,  drawn  through  the 
centre  of  the  earth  parallel  to  MS,  meets 
the  sphere.  The  position  of  the  axis  of 
the  cone  will  be  determined  by  the  right 
ascension  and  declination  of  the  point  Z. 

In  order  to  determine  the  point  Z,  let  the  positions  of  the  sun 
and  moon  be  expressed  by  rectangular  co-ordinates  (Art.  32),  of 
which  the  axis  of  x  is  the  straight  line  drawn  through  the  centre 
of  the  earth  and  the  equinoctial  points,  the  axis  of  y  the  inter- 
section of  the  planes  of  the  equator  and  solstitial  colure,  and 
the  axis  of  z  the  axis  of  the  equator.  Let  x  be  taken  as  positive 
towards  the  vernal  equinox ;  y  as  positive  towards  the  point  of 
the  equator  whose  right  ascension  is  90° ;  z  as  positive  towards 
the  north. 

Let 

a,  d,  r  =  the  right  ascension,  declination,  and  distance  from 

the  centre  of  the  earth,  respectively,  of  the  moon's 

centre, 
</,  <5'?  r'  =  the  right  ascension,  declination,  and  distance  from 

the  centre  of  the  earth,  respectively,  of  the  sun's 

centre; 

The  co-ordinates  x,  y,  z  will  be,  by  (41), 


442  SOLAR    ECLIPSES. 

Of  the  sun.  Of  the  moon. 

r'  cos  d'  cos  a'  r  cos  3  cos  a 

r'  cos  3'  sin  a!  r  cos  3  sin  a 

r'  sin  3'  r  sin  «5 

JSTow  let  another  system  of  co-ordinates  be  taken  parallel  to  the 
first,  the  centre  of  the  moon  being  the  origin.  The  position  of 
the  su\i  in  this  system  will  be  determined  by  the  right  ascension 
and  declination  of  the  sun  as  seen  from  the  moon ;  that  is,  by 
the  right  ascension  and  declination  of  the  point  Z. 
If  we  put 

a,  d  =  the  right  ascension  and  declination  of  the  point  Z, 
G  =  the  distance  of  the  centres  of  the  sun  and  moon, 

the  co-ordinates  of  the  sun  in  the  new  system  are 

G  cos  d  cos  a 
G  cos  d  sin  a 
G  sin  d 

But  these  co-ordinates  are  evidently  equal  respectively  to  the 
difference  of  the  corresponding  co-ordinates  of  the  sun  and  moon 
in  the  first  system ;  so  that  we  have 

G  cos  d  cos  a  =  r'  cos  df  cos  a'  —  r  cos  3  cos  a 
G  cos  d  sin  a  =  r'  cos  3'  sin  a'  —  r  cos  3  sin  a 
G  sin  d  =  r'  sin  3'  —  r  sin  3 

which  fully  determine  a,  rf,  and  G  in  terms  of  quantities  which 
may  be  derived  from  the  Ephemeris  for  a  given  time. 

But,  as  a  and  d  differ  but  little  from  a/  and  d',  it  is  expedient 
to  put  these  equations  under  the  following  form.  (See  the 
similar  transformation,  Art.  92.) 

G  cos  d  sin  (a  —  or)  =  —  r  cos  3  sin  (a  —  a') 

G  cos  d  cos  (a  —  a')  =  r'  cos  d'  —  r  cos  d  cos  (a  —  a') 

G  sin  d  =  r'  sin  d'  —  r  sin  3 

If  these  are  divided  by  r',  and  we  put 

-  —  -  —  b 

they  become 

g  cos  d  sin  (#  —  a')  =  —  5  cos  3  sin  (a  —  a')  *J 

g  cos  d  cos  (a  —  a')  =  cos  dr  —  b  cos  3  cos  (a  —  a')  V   (479) 

a  sin  d  =  sin  d'  —  b  sin  <>  ) 


FUNDAMENTAL    EQUATIONS.  443 

where  the  second  members,  besides  the  right  ascensions  and 
declinations,  involve  only  the  quantity  6,  which  may  be  expressed 
in  terms  of  the  parallaxes  as  follows  : 

Let 

TT  =  the  moon's  equatorial  horizontal  parallax, 
TT'  =  the  sun's  "  "  " 


then  we  have  (Art.  89) 


r        sn 

~ 


r'        sin  TT 
If,  further, 

r0=  the  sun's  mean  horizontal  parallax, 

and  r'  is  expressed  in  terms  of  the  sun's  mean  distance  from  the 

earth,  we  have,  as  in  (146), 

sin  TT 


and  hence 


sin  TT  =  — j- 


b  =  (480) 

r'  sin  TT 


which  is  the  most  convenient  form  for  computing  ft,  because  r' 
and  r  are  given  in  the  Ephemeris,  and  ;rw  is  a  constant. 

290.  The  equations  (479)  are  rigorously  exact,  but  as  b  is  only 
about  jJs,  and  a  —  a'  at  the  time  of  an  eclipse  cannot  exceed 
1°  43',  a  —  a'  is  a  small  arc  never  exceeding  17",  which  may  be 
found  by  a  brief  approximative  process  with  great  precision- 
The  quotient  of  the  first  equation  divided  by  the  second  gives 

b  cos  ft  sec  dr  sin  (a  —  a') 
tan  (a  —  a')  =  — 


1  —  b  COS  d  sec  df  COS  (a  —  a') 

where  the  denominator  differs  from  unity  by  the  small  quantity 
b  cos  S  sec  d'  cos  (a  —  a') ;  and,  since  d  and  d'  are  nearly  equal, 
this  small  difference  may  be  put  equal  to  6,  and  we  may  then 
write  the  formula  thus:* 

cos  d  sec  8f  (o  —  a') 


1  —  b 


*  Developing  the  formula  for  tan  (a  —  a')  in  series,  we  have 

, 5 cos  fi  sec  rT'sin  (a  —  a')        52  cos2  &  sec2  6 'sin  2  (a  —  a') 

sin  1"  "2  sin  1" 

where  the  second  term  cannot  exceed  0".04,  and  the  third  term  is  altogether  inap- 


444  SOLAK    ECLIPSES. 

If  we  take  cos  (a  —  a/)  =  1  and  cos  (a  —  a')  —  1,  we  have, 
from  the  second  and  third  of  (479), 

g  cos  d  —  cos  df  —  b  cos  d 

g  sin  d  =  sin  dr  —  b  sin  d 
whence 

g  sin  (d  —  d')  =  —  b  sin  (d  —  d') 
g  cos  (d  —  d')  =  1  —b  cos  (5  —  Sr) 

from  which  follows 

j  sin  (d  —  d'} 


tan  (d  —  <$')  = 


1  _  b  Cos  ((5  —  <5' 
or,  nearly,* 


From  the  above  we  also  have,  with  sufficient  precision  for  the 
subsequent  application  of  g,  the  formula 


The  formulae  which  determine  the  point  Z,  together  with  the 
quantity  (r,  will,  therefore,  be 


COS  <5  Sec  d'  (a  —  a') 


l—b 

d  =  8' —(d  —  8') 

l  —  b  ^ 

g  =  1  —  6,  G  =  r'g 


(481) 


and  in  many  cases  it  will  suffice  to  take  the  extremely  simple 

forms 

a  =  a.'  —  b  (a  —  a')  d  =  3'—  b  (d  —  <5') 

291.   To  find  the  distance  of  a  given  place  of  observation  from  the 
axis  of  the  shadow  at  a  given  time. — Let  the  positions  of  the  sun, 

preciable.     The  formula  adopted  in  the  text  is  the  same  as 

a  —  a'  =  —  b  cos  6  sec  6 '  (a  —  a')  (1  —  b}~ l 

—  —  b  cos  6  sec  fT'(a  —  a')  —  62  cos  fi  sec  6'  (a —  «,')  —  &c. 

which,  since  cos  6  sec  <Tmay  in  the  second  term  be  put  equal  to  unity,  differs  from 
the  complete  series  only  by  terms  of  the  third  oi'der.  The  error  of  the  approximate 
formula  is,  therefore,  something  less  than  0".01. 

*  The  error  of  this  formula,  as  can  be  easily  shown,  will  never  exceed  0".088. 


FUNDAMENTAL    EQUATIONS. 


445 


Fig.  41  (bit). 
z 


the  moon,  and  the  observer  be  referred  by  rectangular  co-ordi- 
nates to  three  planes  passing  through  the  centre  of  the  earth,  of 
which  the  plane  of  xy  shall  always  be  at  right  angles  to  the  axis 
of  the  shadow,  and  will  here  be  called  the  principal  plane  of  refer- 
ence. Let  the  plane  of  yz  be  the  plane  of  the  declination  circle 
passing  through  the  point  Z.  The  plane 
of  xz  will,  of  course,  be  at  right  angles 
to  the  other  two. 

The  axis  of  z  will  then  be  the  line  OZ, 
Fig.  41,  drawn  through  the  centre  of  the 
earth  parallel  to  the  axis  of  the  shadow, 
and  will  be  reckoned  as  positive  towards 
Z.  The  axis  of  y  wTill  be  the  intersection, 
0  Y,  of  the  plane  of  the  declination  circle 
through  Z  with  the  principal  plane,  and 
will  be  taken  as  positive  towards  the 

north.  The  axis  of  x  will  be  the  intersection,  OX,  of  the  plane 
of  the  equator  writh  the  principal  plane,  and  will  be  taken  as 
positive  towards  that  point,  X,  whose  right  ascension  is  90°  -f  a. 

Let  M'  and  Sf  be  the  true  places  of  the  moon  and  SULL  upon 
the  celestial  sphere,  P  the  north  pole  ;  then,  if  we  put 

x,  y,  z  =  the  co-ordinates  of  the  moon, 

we  have,  by  (Art.  31), 

x  =  r  cos  M' X 
y  =  r  cos  M '  Y 
z  =  r  cos  M  'Z 

which,  by  the  formulae  of  Spherical  Trigonometry  applied  to  the 
triangles  M'PX,  M'PY,  M'PZ,  become 


x  =  r  cos  8  sin  (a  —  a) 

y  =  r  [sin  8  cos  d  —  cos  3  sin  d  cos  (a  —  a)~\   - 

2  =  r  [sin  d  sin  d  -[-  cos  d  cos  d  cos  (a  —  a)~\ 


or 


x  —  r  cos  d  sin  (a  —  a) 

y  =  r  [sin  (d  —  d)  cos2  J  (a  —  a)  -f  sin  (8  -f  d)  sin2  J  (a  —  a)] 

2  =  r  [cos  (<5  —  d)  cos2  £  (a  —  a)  —  cos  (8  -j-  d)  sin2  £  (a  —  a)] 


(482) 


-(482*) 


and  if  the  equatorial  radius  of  the  earth  is  taken  as  the  unit  of 


446  SOLAR   ELLIPSES. 

r,  x,  y,  2,  we  shall  have  the  value  of  r,  required  in  these  equa- 
tions, by  the  formula 


r  = 

sin?: 

The  co-ordinates  x  and  y  of  the  sun  in  this  system  are  the 
same  as  those  of  the  moon,  and  the  third  co-ordinate  is  z  +  G ; 
but  the  method  of  investigation  which  we  are  here  following 
does  not  require  their  use. 

Now  let 

£,  97,  C  =  the  co-ordinates  of  the  place  of  observation, 
<p  =  the  latitude  of  the  place, 
<p'—  the  reduced  latitude  (Art.  81), 
p  =  the  radius  of  the  terrestrial  spheroid  for  the  lati- 
tude <p, 
fj.  =  the  given  sidereal  time ; 

then,  if  in  Fig.  41  we  had  taken  J/ for  the  place  of  observation, 
M '  would  have  been  the  geocentric  zenith  with  the  right  ascen- 
sion ft  and  declination  y* ',  and,  the  distance  of  the  place  from  the 
origin  being  /?,  we  should  have  found 

£  —  p  cos  (/  sin  (/a  —  a)  "j 

•q  =  p  [sin  <f>'  cos  d  —  cos  <f  sin  d  cos  (/j.  —  a)]          V  (483) 

C  =  p  [sin  <p'  sin  d  -j-  cos  <p'  cos  d  cos  (ji  —  a)]          J 

These  equations,  if  we  determine  A  and  B  by  the  conditions 

A  sin  B  =  p  sin  <f 

A  cos  B  =  p  cos  ?>'  cos  (/z  —  a) 

may  be  computed  under  the  form 

£  =  />  cos  95'  sin  (/*  —  a)  "} 


B  —  d)  V(483*) 

C  =  A  cos  ( J?  —  d)  ) 

The  equations  (482)  might  be  similarly  treated;  but  the  most 
accurate  form  for  their  computation  is  (482*). 

The  quantity  //  —  a  is  the  hour  angle  of  the  point  Z  for  the 
meridian  of  the  given  place.  To  facilitate  its  computation,  it  is 
convenient  to  find  first  its  value  for  the  Greenwich  meridian. 
Thus,  if  we  put  for  any  given  Greenwich  mean  time  T 

//j  —  the  hour  angle  of  the  point  Z&t  the  Greenwich  meridian, 
ID  =  the  longitude  of  the  given  place, 


FUNDAMENTAL    EQUATIONS. 


447 


we  have 


a  =  /   — 


To  find  fa  we  have  only  to  convert  the  Greenwich  mean  time  T 
into  sidereal  time  and  to  subtract  a. 

By  means  of  the  formulae  (482)  and  (483)  the  co-ordinates  of 
the  moon  and  of  the  place  of  observation  can  be  accurately  com- 
puted for  any  given  time.  Now,  the  co-ordinates  x  and  y  of  the 
moon  are  also  those  of  every  point  of  the  axis  of  the  shadow  :  so 
that  if  we  put 

A  =  the  distance  of  the  place  of  observation  from  the  axis 
of  the  shadow, 

we  have,  evidently, 


Fig.  42. 


[The  co-ordinates  z  and  £  have  also  been  found,  as  they  will  be 
required  hereafter.] 

292.  The  distance  A  may  be  determined  under  another  form, 
which  we  shall  hereafter  find  useful.  Let  J/', 
Fig.  42,  be  the  apparent  position  of  the  moon's 
centre  in  the  celestial  sphere  as  seen  from  the 
place  of  observation  ;  P  the  north  pole ;  Z  the 
point  where  the  axis  of  the  cone  of  shadow 
meets  the  sphere,  as  in  Fig.  41 ;  Mv  (71?  the 
projections  of  the  moon's  centre  and  of  the 
place  of  observation  on  the  principal  plane. 
The  distance  ClMl  is  equal  to  J,  and  is  the 
projection  of  the  line  joining  the  place  of 
observation  and  the  moon's  centre.  The  plane  by  which  this 
line  is  projected  contains  the  axis  of  the  cone  of  shadow,  and 
its  intersection  with  the  celestial  sphere  is,  therefore,  a  great 
circle  which  passes  through  Z,  and  of  which  ZM'  is  a  portion. 
Hence  it  follows  that  C\M\  makes  the  same  angle  with  the  axis 
of  y  that  M'Z  makes  with  PZ:  so  that  if  we  draw  CyVand 
parallel  to  the  axes  of  y  and  x  respectively,  and  put 

Q  =  PZM'=NC1M1 


we  have,  from  the  right  triangle  C^M 

J  sin  Q  =  x  — 


(* 


the  sum  of  the  squares  of  which  gives  again  the  formula  (484). 


448  SOLAR    ECLIPSES. 

293.  To  find  the  radius  of  the  shadow  on  the  principal  plane,  or  on 
any  given  plane  parallel  to  the  principal  plane. — This  radius  is  evi- 
dently equal  to  the  distance  of  the  vertex  of  the  cone  of  shadow 
from  the  given  plane,  multiplied  by  the  tangent  of  the  angle  of 
the  cone.  In  Figs.  39  and  40,  p.  440,  let  JEF  be  the  radius  of 
the  shadow  on  the  principal  plane,  CD  the  radius  on  a  parallel 
plane  drawn  through  0.  Let 

H  =  the  apparent  semidiameter  of  the  sun  at  its  mean  dis- 
tance, 
k  =  the  ratio  of  the  moon's  radius  to  the  earth's  equatorial 

radius, 

/  =  the  angle  of  the  cone  =  EVF, 

c  =  the  distance  of  the  vertex  of  the  cone  above  the  princi- 
pal plane  =  VF, 

C  =  the  distance  of  the  given  parallel  plane  above  the  prin- 
cipal plane  =  DF, 

I  =  the  radius  of  the  shadow  on  the  principal  plane  =  EF, 
L  =  the  radius  of  the  shadow  on  the  parallel  plane  =  CD. 

If  the  mean  distance  of  the   sun   from   the    earth  is  taken  as 

unity,  we  have 

the  earth's  radius  =  sin  TTO, 

the  moon's  radius  =  k  sin  TTO  =  MB, 

the  sun's  radius     =  sin  H    =  SA, 

and,  remembering  that  G  ==  r'g  found  by  (481)  is  the  distance 
MS,  we  easily  deduce  from  the  figures 

sin  If  -f-  k  sin  ;:„ 

8in/=  -  —^  (486) 

r'g 

in  which  the  upper  sign  corresponds  to  the  penumbral  and  the 
lower  to  the  umbral  cone. 

The  numerator  of  this  expression  involves  only  constant  quan- 
tities. According  to  BESSEL,  H=  959". 788 ;  ENCKE  found 
TTO  ==  8".57116;  and  the  value  of  k,  found  by  BURCKHARDT  from 
eclipses  and  occultations,  is  k  =  0.27227;*  whence  we  have 

log  [sin  H  -f  k  sin  TTO]  =  7.6688033  for  exterior  contacts, 
log  [sin  H  —  k  sin  TTO]  =  7.6666913  for  interior  contacts. 

*  The  value  of  k  here  adopted  is  precisely  that  which  the  more  recent  investiga- 
tion of  OUDEMANS  (Astron.  Nach.,  Vol.  LI.  p.  30)  gives  for  eclipses  of  the  sun. 
For  occultations,  a  slightly  increased  value  seems  to  be  required. 


FUNDAMENTAL    EQUATIONS.  449 

Now,  taking  the  earth's  equatorial  radius  as  unity,  we  have 

k 


sin/ 

MF  =  z    (Art.  291) 
and  hence 

c  =  z  ±  —  (487) 

sin/ 

the  upper  sign  being  used  for  the  penumbra  and  the  lower  for 
the  umbra. 

We  have,  then, 

I  =  c  tan/  =  z  tan  /  ±  k  sec  /  )     ,4gg, 

L  =  (c  —  C)  tan  f=l  —  C  tan  /  j 


For  the  penumbra!  cone,  c  —  f  is  always  positive,  and  there- 
fore L  is  positive  also. 

For  the  umbral  cone,  c  —  £  is  negative  when  the  vertex  of 
the  cone  falls  below  the  plane  of  the  observer,  and  in  this  case 
we  have  total  eclipse  :  therefore  for  the  case  of  total  eclipse  we 
shall  have  L  =  (c  —  £)  tan  /a  negative  quantity.  It  is  usual  to 
regard  the  radius  of  the  shadow  as  a  positive  quantity,  and 
therefore  to  change  its  sign  for  this  case  ;  but  the  analytical  dis- 
cussion of  our  equations  will  be  more  general  if  we  preserve 
the  negative  sign  of  L  as  the  characteristic  of  total  eclipse. 

When  the  vertex  of  the  umbral  cone  falls  above  the  plane  of 
the  observer,  L  is  positive,  and  we  have  the  case  of  annular 
eclipse. 

For  brevity  we  shall  put 

i  =  tan/  ^ 

l=.ic  >    (489) 

Ii=l  —  i:  ) 

294.  The  analytical  expression  of  the  condition  of  beginning  or 
ending  of  eclipse  is 

A=Ii 
or,  by  (484)  and  (489), 

(x  -  *)'  +  (y  -  ,)«  =(l-  i:y  (490) 

It  is  convenient,  however,  to  substitute  the  two  equations 
(485)  for  this  single  one,  after  putting  L  for  J,  so  that 


C4' 

VOL.  I.—  29 


450  SOLAR   ECLIPSES. 

may  be  taken  as  the  conditions  which  determine  the  beginning 
or  ending  of  an  eclipse  at  a  given  place. 

The  equation  (490),  which  is  only  expressed  in  a  different  form 
by  (491),  is  to  be  regarded  as  the  fundamental  equation  of  the 
theory  of  eclipses. 

295.  By  Art.  292,  so  long  as  A  is  regarded  as  a  positive  quan- 
tity, Q  is  the  position  angle  of  the  moon's  centre  at  the  point  Z; 
and  since  the  arc  joining  the  point  Z  and  the  centre  of  the  moon 
also  passes  through  the  centre  of  the  sun,  Q  is  the  common 
position  angle  of  both  bodies. 

Again,  since  in  the  case  of  a  contact  of  the  limbs  the  arc 
joining  the  centres  passes  through  the  point  of  contact,  Q 
will  also  be  the  position  angle  of  this  point  when  all  three 
points — sun's  centre,  moon's  centre,  and  point  of  contact — lie 
on  the  same  side  of  Z.  In  the  case  of  total  eclipse,  however, 
the  point  of  contact  and  the  moon's  centre  evidently  lie  on 
opposite  sides  of  the  point  Z;  and  if  I  —  if  in  (490)  were  a 
positive  quantity,  the  angle  Q  which  would  satisfy  these  equa- 
tions would  still  be  the  position  angle  of  the  moon's  centre,  but 
would  differ  180°  from  the  position  angle  of  the  point  of  con- 
tact. But,  since  we  shall  preserve  the  negative  sign  of  I  —  zf 
for  total  eclipse  (Art.  293),  (and  thereby  give  Q  values  which 
differ  180°  from  those  which  follow  from  a  positive  value),  the 
angle  Q  will  in  all  cases  be  the  position  angle  of  the  point  of  contact. 

296.  The  quantities  a,  d,  x,  y,  £,  and  i  may  be  computed  by 
the  formula  (480),  (481),  (482),  (486),  (487),  (488),  for  any  given 
time  at  the  first  meridian,  since  they  are  all  independent  of  the 
place  of  observation.     In  order  to*  facilitate  the  application  of 
the  equations  (490)  and  (491),  it  is  therefore  expedient  to  com- 
pute  these  general  quantities  for   several   equidistant   instants 
preceding  and  following  the  time  of  conjunction  of  the  sun  and 
moon,  and  to  arrange  them  in  tables  from  which  their  values 
for  any  time  may  be  readily  found  by  interpolation. 

The  quantities  x  and  y  do  not  vary  uniformly  ;  and  in  order  to 
obtain  their  values  with  accuracy  from  the  tables  for  any  time, 
we  should  employ  the  second  and  even  the  third  differences  in 
the  interpolation.  This  is  effected  in  the  most  simple  manner 
by  the  following  process.  Let  the  times  for  which  x  and  y  have 
been  computed  be  denoted  by  TQ  —  2%  TQ  —  1*,  T»  TQ  +  l\ 


FUNDAMENTAL    EQUATIONS. 


451 


T0  -f-  2*,  the  interval  being  one  hour  of  mean  time ;  and  let  the 
values  of  x  and  y  for  these  times  be  denoted  by  x_2,  x_i,  &c., 
t/_2,  y_i,  £c.  Let  the  mean  hourly  changes  of  x  and  y  from  the 
epoch  7"0  to  any  time  T  ==  TQ  -j-  r  be  denoted  by  x'  and  y' .  Then 
the  values  of  x'  and  #'  for  the  instants  7"0  —  2*,  T0  —  1*,  &c.  will 
be  formed  as  in  the  following  scheme,  where  c  denotes  the  third 
difference  of  the  values  of  x  as  found  from  the  series  x_2,  x_i,  &c. 
according  to  the  form  in  Art.  69,  and  the  difference  for  the 
instant  T0  is  found  by  the  first  formula  of  (77).  The  form  for 
computing  y'  is  the  same.  , 


Time. 


77         o 

a          * 


T.+  V 


3  (X-2  — 


If  then  we  require  x  and  y  for  a  time  T  =  TQ  -\-  r,  we  take 
re'  and  j/'  from  the  table  for  this  time,  and  we  have 

x  =  x0  +  x'r 

y  =  </0  +  /T 

297.  EXAMPLE. — Compute  the  elements  of  the  solar  eclipse  of 
July  18,  1860. 

The  mean  Greenwich  time  of  conjunction  of  the  sun  and 
moon  in  right  ascension  is  July  18,  2h  Sm  56*.  The  computation 
of  the  elements  will  therefore  be  made  for  the  Greenwich  hours 
0,  1,  2,  3,  4,  and  5.  For  these  hours  we  take  the  following 
quantities  from  the  American  Ephemeris  : 

For  the  Moon. 


Greenwich  mean 
time. 

a 

6 

7T 

July  18,  0* 

116°  44'  24".30 

21°  52'  20".3 

59'  45".80 

1 

117  21  59  .10 

42  32  .8 

47  .13 

2 

117  59  30  .45  . 

32  36  .4 

48  .44 

3 

118  36  58  .35 

22  31  .2 

49  .72 

4 

119  14  22  .65 

12  17  .2 

50  .98 

5 

119  51  43  .35 

1  54  .6 

52  .22 

452 


SOLAR    ECLIPSES. 
For  the  Sun. 


Greenwich  mean 
time. 

a' 

6' 

i 

logr' 

July  18,  0* 

117°  59'  41".85 

20°  57'  56".20 

0.0069675 

1 

118      2   12  .50 

57  29  .42 

61 

2 

118      4  43  .14 

57     2  .60 

47 

3 

118      7  13  .77 

56  35  .75 

33 

4 

118      9  44  .39 

56     8  .86 

19 

5 

118*  12  15  .00 

55  41  .94 

05 

The  formulae  to  be  employed  will  be  here  recapitulated,  for 
convenient  reference. 

I.  For  the  elements  of  the  point  Z: 


b  = 


sin 


r'  sin  TT 


log  sin  XQ=  5.61894 


a  =  a, — 


1  —  6 


cos  3  sec  3'  (a  —  a')     or,  nearly,     a  =  of  —  b  (a  —  a') 


d  =  8r  — 


l  —  b" 
9  =  1  —  & 

II.  The  moon's  co-ordinates  : 
1 


r  = 


sin 


x  =  r  cos  8  sin  (a  —  a) 

y  =  r  sin  (3  —  d)  cos2  £  (a  —  a)  -f-  r  sin  (5  -f  d}  sin2  J  (a  —  a)- 

2  =  r  cos  (<?  —  d)  cos2  J  (a  —  a)  —  r  cos  (<J  -f  d)  sin2  J  (a  —  a) 

III.  The  angle  of  the  cone  of  shadow  and  the  radius  of  the 
shadow  : 


For  penumbra:  or  exterior  contacts. 

[7.668803] 


sin/  = 


For  umbra:  or  interior  contacts 

[7.666691] 
Bin/=  L— 


sin/ 


log  k  =  9.435000,     c  =  2 : 


sin/ 


i  =  tan  / 


=  tan  / 


FUNDAMENTAL    EQUATIONS. 


453 


IV.  The  values  of  a,  d,  x,  y,  log  i,  and  I,  will  then  be  tabulated 
and  the  differences  xf  and  y'  formed  according  to  Art. 

I  give  the  computation  for   the  three  hours  1*,  2 
in  extenso. 


I.  Elements  of  the  point  Z. 


81 


1* 

2* 

3* 

a  —  a' 

—  0°  40'  13".  40 

—  0°  5'  12".  69 

+0°  29'  44".  58 

6—6' 

+   45  3  .38 

+   35  33  .80 

+   25  55  .45 

log  cosec  TT  =  log  r 

1.7596999 

1.7595414 

1.7593865 

ar.  co.  log  r' 

9.9930339 

9.9930353 

9.9930367 

Constant  log  sin  TTQ 

5.61894 

log  b 

7.37167 

7.37152 

7.37136 

ar.  co.  log  (1  —  b) 

0.001023 

0.001023 

0.001022 

log  cos  6 

9.96805 

9.96855^ 

9.96905 

log  sec  tV 

0.02973 

0.02970 

0.02968 

log  (a  —  a') 

n3.  38263 

«2.  49511 

3.25154 

log  (a  —  a') 

0.75310 

9.86590 

nO.  62265 

a  —  a' 

+      5".  66 

+      0".73 

4".  19 

) 

7.37269 

7.37254 

7.37238 

log  (6  -  6') 

3.43191 

3.32915 

3.19185 

log  (d—6') 

nO.  80460 

nO.70169 

wO.  56423 

d—6' 

6".  38 

5".  03 

3".  67 

a 

118°  2'  18".16 

118°  4'43".87 

118°  7'  9".  58 

d 

20  57  23  .04 

20  56  57  .57 

20  56  32  .08 

log  (1  —  b)  —  log  g 

9.998977 

9.998977 

9.998978 

II.  Co-ordinates  x,  yy  and  z. 


a  —  a 
6  —  d 

6  +  d 

log  sin  (a  —  a) 
log  cos  6 

log  r  cos  6  sin  (a  —  a)  =  log  x 
x 

log  cos2 2 -(a  —  a 
log  sin(d  —  d 

log  (3)  =logrsin  (6  —  rf)cos2  £  (a  —  a 
log  sin2  £  (a  —  a) 
log  sin(<J-f-d) 

fog  (4)  =log  r  sin  (6  -f-  d)  sin2  £  (a  —  a) 

(3) 

(4) 

(3)  +  (4)  y 

log  cos  (6  —  d) 
c?)cos2£(a — a) 
log  cos  (6  -\-d) 

log  (6)  =logr  cos  (6  -f-  d)sin2 }  (a  —  a) 
log  [(5)  —  (6)]  =  log  z 


-0°  40'  19".06 

—  0°  5'  13".42i+0°  29'  48".77 

+   45  9  .76 

+   35  38  .83 

+   25  59  .12 

42  39  55  .84 

42  29  33  .97 

42  19  3  .28 

n8.0692116 

nl.  181  701  4 

7.9381239 

9.9680502 

9.9685481 

9.9690490 

n9.  79696  17 

«8.9097909 

9.6665594 

—0.626559 

—0.081244 

0.464044 

9.9999850 

9.9999998 

9.9999920 

8.1184932 

8.0157434 

7.8784502 

9.8781781 

9.7752846 

9.6378287 

5.5363780 

3.7613394 

5.2741910 

9.8310485 

9.8296235 

9.8281695 

7.1271264 

5.3405043 

6.8617470 

+0.755402 

+0.596053 

+0.434329 

+0.001340 

+0.000022 

+0.000727 

-f  0.756742 

+0.596075 

+0.435056 

9.9990625 

9.9999766 

9.9999876 

1.7596474 

1.7595178 

1.7593661 

9.8664780 

9.8676822 

9.8688939 

7.1625559 

5.3885630 

6.9024714 

1.7596364 

1.7595176 

1.7593601 

454  SOLAR    ECLIPSES. 

HE.  Log  i  and  I,  for  exterior  contacts.  [Constant  log  =  7.668803] 


1A 

2, 

3* 

log  r'# 

0.005943 

0.005942 

0.005941 

Const.  - 

log  r'g  •==.  log  sin/ 

7.662860 

7.662861 

7.662862 

log  sec/ 

O.C00005 

log  &  cosec/ 

1.772140 

1.772139 

1.772138 

log  [z  + 

k  cosec  /]  =  log  c 

2.066963 

2.066904 

2.066826 

log  tan/ 

=  log  » 

7.662865 

7.662866 

7.662867 

log  ic 

=  log  Z 

9.729828 

9.729770 

9.729693 

Z 

0.536819 

0.536747 

0.536652 

Log  i  and  I 

for  interior  contacts.     [Constant  log  =  7.666691] 

Const.  — 

log  r'g  —  log  sin/      7.660748             7.660749             7.660750 

log  sec/ 

0.000005 

log  k  cosec/ 

1.774252 

1.774251 

1.774250 

log  0  —  i 

fc  cosec/]  =  log  c 

nO.  293985 

nO.297413 

nO.301919 

log  tan/ 

—  log  * 

7.660753 

7.660754 

7  660755 

log  ic 

=  log  I 

w7.9o4738 

n7.958167 

n7.962674 

I 

—0.009010 

—0.009082 

—0.009176 

IV.  The  computation  being  made  for  the  other  hours  in  the 
same  manner,  the  results  are  collected  in  the  following  tables. 


Exterior  Contacts. 

Interior  Contacts. 

I 

log* 

I 

logf 

f 

0* 

117°59'52".44 

20°  57'  48".50 

0.536867 

7.662864 

—  0.008960 

7.660752 

1 

118      2  18  .16 

57  23  .04 

0.536819 

65 

0.009010 

53 

2 

4  43  .87 

56  57  .57 

0.536747 

66 

0.009082 

54 

3 

7    9  .58 

56  32  .08 

0.536652 

67 

0.009176 

55 

4 

9  35  .27 

56     6  .58 

0.536533 

68 

0.009293 

56 

5 

12    0  .95 

55  41  .06 

0.536391 

69 

0.009434 

57 

X 

A» 

A* 

A3 

—45 

—60 

—75 

y 

A, 

A2 

^3 

+  17 
+  19 

+20 

0* 
1 
2 

3 
4 
5 

—  1.171856 
—  0.626559 
—  0.081244 
+  0.464044 
+  1.009245 
+  1.554284 

+  0.545297 
0.545315 
0.545288 
0.545201 
0.545039 

•f  18 
—  27 
—  87 
—162 

1+0.917040 
'+0.756742 
!+  0.596075 
i+  0.435056 
!+  0.273704 
+  0.112039 

—  0.160298 
—  0.160667 
—  0.161019 
—  0.161352 
—  0.161665 

—369 
—352 
—333 
—313 

For  the  values  of  the  hourly  differences  of  x  and 
from  the  above,  by  Art.  296, 


we 


find 


FUNDAMENTAL    EQUATIONS. 


455 


x' 

log*' 

y' 

logy'    >j 

0* 

0.545306 

9.736640 

-  0.160483 

W9.205429 

1 

0.545315 

648 

-0.160607 

5927 

TO  =  2 

0.545310 

644 

-  0.160846 

6410 

3 

0.545288 

626 

-0.161019 

6877 

4 

0.545245 

592 

-0.161186 

7327 

5 

0.545176 

537 

—  0.161345 

7756 

and  for  any  given  time  T  =  TQ  H~  r,  we  have 

#  =  —  0.081244  +  afr 
y  =  -f  0.596075  -f  y'r 


}    (492) 


Finally,  to  facilitate  the  computation  of  the  hour  angle 
H  —  a  =•  fa  —  co  (Art.  291),  we  prepare  the  values  of  /^  for  each 
of  the  Greenwich  hours.  Thus,  for  T  =  1A,  we  have 


4*. 03 

9  .86 


From  the  Ephemeris,  July  18,  1860, 

Sid.  time  at  mean  noon  =     7*  46* 

Sid.  equivalent  of  1*  mean  t.   =     1      0 

Greenwich  sid.  time  =     8    46   13  .89 

"  «       «       in  arc,   =131°  33'  28".35 

a   =118      2   18  .16 


=   13    31   10  .19 


Thus  we  form  the  following  table,  to  which  is  also  added  for 
future  use  the  value  of  the  logarithm  of 

//  =  the  hourly  difference  of  ^  in  parts  of  the  radius; 


ft 

Hourly  diff. 

0A 

358°  31'     8".0 

1 

13     31    10  .2 

2 

28     31    12  .3 

54002M5 

3 

43     31    14  .4 

4 

58     31    16  .6 

5 

73     31    18  .7 

log  fir  =  log  54002M5  sin  1" 
=  9.417986 


I  proceed  to  consider  the  principal  problems  relating  to  the 
general  prediction  of  eclipses,  in  which  the  preceding  results 
will  be  applied. 


456  SOLAR   ECLIPSES. 

Outline  of  the  Shadow  on  the  Surface  of  the  Earth. 

298.  To  find  the  outline  of  the  moon's  shadow  upon  the  earth  at  a 
given  time. — This  outline  is  the  intersection  of  the  cone  of  shadow 
with  the  earth's  surface ;  or,  it  is  the  curve  on  the  surface  of  the 
earth  from  every  point  of  which  a  contact  of  the  sun's  and 
moon's  limbs  may  be  observed  at  the  given  time.  Let 

T =  the  given  time  reckoned  at  the  first  meridian, 

and  let  a,  d,  x,  y,  I,  and  log  i  be  taken  from  the  general  tables 
of  the  eclipse  for  this  time.  Then  the  co-ordinates  £,  ^,  f  of  any 
place  at  which  a  contact  may  be  observed  at  the  given  time  must 
satisfy  the  conditions  (491), 

tfl^co^elly-*  }    (498) 

Let 

#  =  the  hour  angle  of  the  point  Z, 
w=  the  west  longitude  of  the  place; 

then  we  have 

#  =  fj.  —  a  =  p.±  —  at 

and  the  equations  (483)  become 

£  =  p  cos  <p'  si  n  #  ^ 

y  =  p  sin  <p'  cos  d  —  p  cos  <?'  sin  d  cos  &  V    (494) 

C  =  p  sin  <p'  sin  d  -{-  p  cos  <p'  cos  d  cos  #  J 

The  five  equations  in  (493)  and  (494)  involve  the  six  variables 
£,  37,  £,  <p'9  $,  and  §,  any  one  of  which  may  be  assumed  arbi- 
trarily (excluding,  of  course,  assumed  values  that  give  impossible 
or  imaginary  results) ;  then  for  each  assumed  value  qf  the  arbi- 
trary quantity  we  shall  have  five  equations,  which  fully  deter- 
mine five  unknown  quantities,  and  thereby  one  point  of  the  re- 
quired curve.  I  shall  take  Q  as  the  arbitrary  variable. 

In  the  present  form  of  the  equations  (494),  they  involve  the 
unknown  quantity  />,  which  being  dependent  upon  <pf  cannot  be 
determined  until  the  latter  is  found.  This  seems  to  involve  the 
necessity  of  at  first  neglecting  the  compression  of  the  earth,  by 
putting  f)  —  1,  and  after  an  approximate  value  of  <p'  has  been 
found,  and  thereby  also  the  value  of  />,  repeating  the  computation. 
But,  by  a  simple  transformation  given  by  BESSEL,  this  double 
computation  is  rendered  unnecessary,  and  the  compression  of 


OUTLINE    OF   THE    SHADOW.  457 

the  earth  is  taken  into  account  from  the  beginning.     If  <p  is  the 
geographical  latitude,  we  have  (Art.  82) 

cos  <p  sin  <p  (\  —  ee) 

' ' 


—  eesm'yj  I/A1  —  ee  sin-1 

in  which 

log  ee  =  7.824409  log  j/(l  —  ee)  =  9.9985458 

If  we  take  a  new  variable  yl9  such  that 

cos  <p 

cos  y.  — 

l/(l  —  ee  sm2  9?) 
we  shall  have 

sin  <p  T/(l  —  ee) 
sin  ?,  =  T/(l  —  cos2  <O  =  — ^- 

—  ee  sm2  <p\ 


or 

cos  <>  =     cos 


ee)  sin  <pl  =  />  sin  ?>' 
tan  c?. 


tan      = 


Hence  the  equations  (494)  become 

£  =  cos  <p  l  sin  # 

iy  =  sin  ^  cos  d  |/(1  —  ee)  —  cos  ^  sin  ^  cos  ft 

C  =  sin  ^  sin  <Z  -|/(1  —  ee)  +  cos  ^j  cos  d  cos  # 

Put 


sin  dl  =  sin  <Z  pa  sin  ^2  =  sin  d  j/(l  —  ee) 

cos  dl  =  cos  ^  |/(1  —  ee)         ^3  cos  d.2  =  cos  ^ 


The  quantities  /?„  ^,  />2,  c?2,  may  be  computed  for  the  same  times 
as  the  other  quantities  in  the  tables  of  the  eclipse,  and  hence 
obtained  by  interpolation  for  the  given  time.  The  factors 
pl  and  f>2  will  be  sensibly  constant  for  the  whole  eclipse.  "We 
now  have 

$  =  cos  <pl  sin  # 

TTJ  —  pi  sin  yl  cos  dl  —  pl  cos  ^  sin  dl  cos  t9 
C  =  py  sin  ^>t  sin  d2  -f-  /o2  cos  ^x  cos  d2  cos  t? 
Let  us  put 


And  assume  fn  so  that 

e2  +  ^  +  C,3  -  1  (496) 


458  SOLAR    ECLIPSES. 

or,,  which  is  equivalent,  let  us  take  the  system 

£  =  cos  yl  sin  ft  \ 

T]I  =  sin  <pl  cos  dl  —  cos  <pl  sin  dl  cos  ft  V    (497) 

C,  =  sin  §?j  sin  d^  -f-  cos  <p l  cos  dl  cos  ft  ) 

The  quantity  f  x  differs  so  little  from  f  that  we  may  in  practice 
substitute  one  for  the  other  in  the  small  term  if ;  but  if  theo- 
retical accuracy  is  desired  we  can  readily  find  f  when  ft  is 
known ;  for  the  second  and  third  of  (497)  give 

cos  <pl  cos  ft  =  —  yl  sin  dl  -f-  Ct  cos  dl 
sin  ^x  =       ^x  cos  dl  -\-  Ct  sin  ^ 

which  substituted  in  the  value  of  f  give 

C  =  p2  Cj  cos  (^  —  6?2)  —  ft  ^j  sin  (dl  —  d.2)  (498) 

Our  problem  now  takes  the  following  form.     We  have  first 
the  three  equations 

(?  —  iCj)  sin  Q  =  x  —  £  ^| 

(z  _  j-y  cos  (2  =  y  —  ft  ^  r  ^4") 


which  for  each  assumed  value  of  Q  determine  £,  i^,  and  £r    Then 

we  have 

cos  <pl  sin  ^  =       £ 

cos  ^  cos  #  =  —  ^j  sin  ^  -f  C,  cos  dl  >    (500) 

sin^^r       ^  cos  dl  +  Cj  sin  dl 


1 

> 
) 


which  determine  <pl  and  ??.     Then  the  latitude  and  longitude  of 
a  point  of  the  required  outline  are  found  by  the  equations 

tan  9  ==     tan^  «,  =  ^  -  «  (501) 


To  solve  (499),  let  /9  and  y  be  found  by  the  equations 

sin  /?  sin  ?  =  %  —  I  sin  Q  =  d     • 

(    rS***^ 

y        I  cos  Q 
sin  /9  cos  Y  =  — —  b 

Pi  Pi 

then  we  have 

£  =  sin  /?  sin  y  -f-  iCj  sin  § 
^  =  sin  /9  cos  ^  -f-  iTj  cos  Q 


(502) 


OUTLINE    OF    THE    SHADOW.  459 

where  we  have  omitted  ^  as  a  divisor  of  the  small  term  ?'£,  cos  §, 
since  we  have  very  nearly  pl  —  1.  Substituting  these  values  in 
the  last  equation  of  (499),  we  find 

t*  =  cos2/?  —  2^  sin  ft  cos  (Q  —  r)  —  (i^ 

Neglecting  the  terms  involving  z2  as  practically  insensible,  this 
gives 

Cx=  ±  [cos  /?  —  i  sin  £  cos  (Q  —  r^j] 

In  order  to  remove  the  ambiguity  of  the  double  sign,  let  us  put 

Z  =  the  zenith  distance  of  the  point  Z  (Art.  289)  ; 
then,  since  &  =  /ut  —  a  is  the  hour  angle  of  this  point,  we  have 

cos  Z  =  sin  <p  sin  d  -f-  cos  y  cos  d  cos  # 
which  by  means  of  the  preceding  equations  is  reduced  to 

(503) 


sin  ^ 

Hence  cos  Z  and  £\  have  the  same  sign. 

But,  in  order  that  the  eclipse  may  be  visible  from  a  point  on 
the  earth's  surface,  we  must,  in  general,  have  J^less  than  90°; 
that  is,  cos  Z  must  be  positive,  and  therefore  £t  must  be  taken 
only  with  the  positive  sign.  The  negative  sign  would  give  a 
second  point  on  the  surface  of  the  earth  from  which,  if  the  earth 
were  not  opaque,  the  same  phase  of  the  eclipse  would  also  be 
observed  at  the  given  time.  In  fact,  every  element  of  the  cone 
of  shadow  which  intersects  the  earth's  surface  at  all,  intersects 
it  in  two  points,  and  our  solution  gives  both  points. 

If  we  put 

>  =  iCQ8(C-r) 

sin  1" 
we  have 

C,  =  cos  /5  —  sin  p  sin  e 

or,  with  sufficient  accuracy, 

C1=cos(j9  +  e)  (505) 

Thus,  j9  and  f  being  determined  by  (502),  £:  is  determined  by 
(504)  and  (505)  :  hence  also  £  and  ^  by  the  equations 


,  =  *  +  iC,  cos  Q 


i 

/  <• 


460  SOLAR   ECLIPSES. 

The  problem  is,  therefore,  fully  resolved ;  but,  for  the  conve- 
nience of  logarithmic  computation,  let  c  and  C  be  determined 
by  the  equations 

C8m^   =ll  1(507) 

c  cos  C  =  Ct  / 

then  the  equations  (500)  become 

cos  <p l  sin  $  =  £  ^ 

cos  ^  cos  #  =  c  cos  (<7  -j-  dj)  V  (508) 

sin  ^t=  c  sin  (C  -j-  dj)  ) 

The  curve  thus  determined  will  be  the  intersection  of  the 
penumbral  cone,  or  that  of  the  umbral  cone,  with  the  earth's 
surface,  according  as  we  employ  the  value  of  I  for  the  one  or  the 
other. 

299.  The  above  solution  is  direct,  though  theoretically  but 
approximate,  since  we  have  neglected  terms  of  the  order  of  i2. 
It  can,  however,  readily  be  made  quite  exact  as  follows.  "We 
have,  by  substituting  the  values  of  f  t  and  ^1  in  (498),  and  neg- 
lecting the  term  involving  the  product  ism(dl — flQ,  which  is 
of  the  same  order  as  i2, 

C  =  /?2  cos  (/?  -f~  0  —  P-i  sin  ft  cos  Y  s^n  (^i  —  ^2) 

and,  putting 

e'  =  (dl  —  d^)  cos  Y 

we  have,  within  terms  of  the  order  i2, 

C  =  f>2  cos  (/?  -f  £  -f  e')  (509) 

The  substitution  of  this  value  of  f  in  the  term  if  involves  only 
an  error  of  the  order  i3,  which  is  altogether  insensible.  The 
exact  solution  of  the  problem  is,  therefore,  as  follows.  Find  /9 
and  f  for  each  assumed  value  of  §,  by  the  equations 

sin  /?  sin  f  =  x  —  I  sin  Q  =  a 
=  y      I  cosg  __  b 


then  e  and  e'  by  the  equations 


i  cos  (0  —  r} 

e  =  —  «  -  5^  e'=  (^  _  ^  )  cos 

sin  1"  ^  i 


OUTLINE    OF    THE    SHADOW.  461 

Find  /9'  and  f  by  the  equations 

sin  /?'  Bin  /  =  a  -f-  ip.2  cos  (ft  -}-  e  -j-  e')  sin  Q  =  g 

f  lp2  COS  (/?  -j-  e  -f  e')  COS  # 

sm  /?'  cos  /=  £>-}--  =  ^i 

then  we  have,  rigorously, 

5V=cp*/?< 

and  these  values  of  £,  ^1?  and  ^l  may  then  be  substituted  in  (500), 
which  can  be  adapted  for  logarithmic  computation  as  before.* 

300.  It  remains  to  be  determined  whether  the  eclipse  is  begin- 
ning or  ending  at  the  places  thus  found.  A  point  on  the  earth's 
surface  which  at  a  given  time  T  is  upon  the  surface  of  the  cone 
of  shadow  will  at  the  next  consecutive  instant  T  -f  dT  be 
ieithin  or  without  the  cone  according  as  the  eclipse  is  beginning  or 
ending  at  the  time  T;  the  former  or  the  latter,  according  as  the 
distance  A  =  i/[(x  —  c)2  +  (y  —  ^)2]  becomes  at  the  time  T  -f  dl 
less  or  greater  than  the  radius  of  the  shadow  I  —  /£.  In  the  case 
of  total  eclipse  I  —  if  is  a  negative  quantity,  but  by  comparing 
J2  with  (I  —  i£f  we  shall  obtain  the  required  criterion  for  all 
cases ;  and,  therefore,  the  criterion  of  beginning  or  ending,  either 
of  partial  or  of  total  eclipse,  will  be  the  negative  or  positive  value 
of  the  differential  coefficient,  relatively  to  the  time,  of  the 
quantity 

(x  -  O2  +  (y  -  7)1  -  (I  -  *0' 

or  the  negative  or  positive  value  of  the  quantity 

^_«\+(,/_,)(^_^\_a_!>)(^_^ 

dT      dTJ  u\dT      dT]  \dT         dT 

*  In  this  problem,  as  well  as  in  most  of  the  subsequent  ones,  I  have  not  followed 
BESSEL'S  methods  of  solution,  which,  being  mathematically  rigorous,  though  as 
simple  as  such  m3thods  can  possibly  be,  are  too  laborious  for  the  practical  purposes 
of  mere  prediction.  As  a  refined  and  exhaustive  disquisition  upon  the  whole  theory, 
BESSEL'S  Analyse  der  Finsternisse,  in  his  Astronomische  Untersuchungcn,  stands  alone. 
On  the  other  hand,  the  approximate  solutions  heretofore  in  common  use  are  mostly 
quite  imperfect ;  the  compression  of  the  earth,  as  well  as  the  augmentation  of  the 
moon's  semidiameter,  being  neglected,  or  only  taken  into  account  by  repeating  the 
whole  computation,  which  renders  them  as  laborious  as  a  rigorous  and  direct  method. 
I  have  endeavored  to  remedy  this,  by  so  arranging  the  successive  approximations, 
when  these  are  necessary,  that  only  a  small  part  of  the  whole  computation  is  to  be 
repeated,  and  by  taking  the  compression  of  the  earth  into  account,  in  all  cases,  from 
the  commencement  of  the  computation.  In  this  manner,  even  the  first  approxima- 
tions by  my  method  are  rendered  more  accurate  than  the  common  methods. 


462  .      SOLAR    ECLIPSES. 

where  we  omit  the  insensible  variation  of  L     For  brevity,  let  us 
write  x',  ?/',  &c.  for  ~r^,  "Ir"1  ^c'  an^  Denote  ^e  above  quantity 

by  P;  then,  after  substituting  the  values  of  x  —  £  =  (l  —  1£)  sin  §, 
y  —  7j  =  (I  —  ir)  cos  §,  we  have 


P=L  [(*'  —  £')  sin  Q  +  (yf  —  ,')  cos  §  —  (?'—  fC')] 
in  which  L=l  —  if.     If  we  put 

P'  =  (*'  -  £')  sin  Q  -f  (y'  -  ,  ')  cos  Q  -  (I'  -  tO        (510) 

we  shall  have 

P  =  LP' 

The  quantity  P  will  be  positive  or  negative  according  as  L  and 
P'  have  like  signs  or  different  signs. 

For  exterior  contacts,  and  for  interior  contacts  in  annular 
eclipse,  L  is  positive  (Art.  293),  and  hence  for  these  cases  the  eclipse 
is  beginning  or  ending  according  as  P'  is  negative  or  positive;  but  for 
total  eclipse,  L  being  negative,  we  have  beginning  or  ending 
according  as  P'  is  positive  or  negative. 

We  must  now  develop  the  quantity  P'  '.  Taking  one  hour  as 
the  unit  of  time,  x',  y'  ,  I'  ,  £',  r/,  £',  will  denote  the  hourly  changes 
of  the  several  quantities.  The  first  three  of  these  may  be 
derived  from  the  general  tables  of  the  eclipse  for  the  given  time; 
but  £',  37',  f  '  are  obtained  by  differentiating  the  equations  (494), 
in  which  the  latitude  and  longitude  of  the  point  on  the  earth's 
surface  are  to  be  taken  as  constant.  Since  &  =  fa  —  o>,  we  shall 

have  -j-^  =    -^  ;  and  hence,  putting 

» 

.       dfi.    .  dd    .      „ 

fif  =  —  --  sin  V  df  =  —  sm  1" 

dT  dT 
we  find 

£'  =  n'p  cos  /cos  »7  =  //  (  —  T]  sin  d  -j-  C  cos  rf) 

=  A*'  [  —  ?/  sin  6?  -}-  C  cos  d  -f-  (Z  —  i'C)  sin  d  cos  §] 
TJ'  =  p'  $  sin  rf  —  d'Z 

=  //  [a:  sin  d  —  (?  —  i'C)  sin  d  sin  <2]  —  d'Z 

£'=  —  //f  COS  6?  -f  (Try 

=  ^'  [  —  #  cos  d  -f-  (I  —  iT)  cos  c?  sin  Q]  -\-  d'  [y  —  (I  —  i'C)  cos  §] 


OUTLINE    OF    THE    SHADOW.  463 


Substituting  these  values  in  (510),  and  neglecting  terms  involving 
i2  and  id!  as  insensible,  we  have 

P'  =  a'  —  bf  cos  Q  -f-  c'  sin  Q  —  C  (//  cos  d  sin  Q  —  d'  cos  Q) 
in  which  af,  bf,  and  c',  denote  the  following  quantities: 

a'  =  —  I'  —  //  ix  cos  d  ^ 

c'  =       x1  -f-  fir  y  sin  6?  -f-  ji'  z7  cos  d  ) 

The  values  of  these  quantities  may  be  computed  for  the  same 
times  as  the  other  quantities  in  the  eclipse  tables,  and  their 
values  for  any  given  time  will  then  be  readily  found  by  interpo- 
lation. For  any  assumed  value  of  Q,  therefore,  and  with  the 
value  of  £  found  by  (509),  the  value  of  P'  may  be  computed,  and 
its  sign  will  determine  whether  the  eclipse  is  beginning  or 
ending.  In  most  cases,  a  mere  inspection  of  the  tabulated  values 
of  a1 ',  6',  and  c',  combined  with  a  consideration  of  the  value  of 
$,  will  suffice  to  determine  the  sign  of  P1 ;  but  when  the  place 
is  near  the  northern  or  southern  limits  of  the  shadow,  an  accu- 
rate computation  of  P'  will  be  necessary ;  and,  since  other  appli- 
cations of  this  quantity  will  be  made  hereafter,  it  will  be  proper 
to  give  it  a  more  convenient  form  for  Idjirithmic  computation. 

Put 

e  sin  E  =  bf  f  sin  F  =  d' 


then  we  have 

P'  ==  a'  -f  e  sin  (Q  —  E)  —  C/  sin  (Q  —  JP)  (513) 

Since  af  and  jF'are  both  very  small  quantities,  and  a  very  precise 
computation  of  P'  will  seldom  be  necessary  when  its  algebraic 
sign  is  alone  required,  it  will  be  sufficient  in  most  cases  to  neglect 
these  quantities,  and  also  to  put  ft  for  £,  and  then  we  shall  have 
the  following  simple  criterion  for  the  case  of  partial  or  annular 
eclipse : 

If    e  sin  (Q  —  E)  <  Cj/  sin  Q,  the  eclipse  is  beginning. 
If    e  sin  (Q  —  E)  >  C4/  sin  Q,  the  eclipse  is  ending. 

For  total  eclipse,  reverse  these  conditions. 

301.  In  order  to  facilitate  the  application  of  the  preceding  as 
well  as  the  subsequent  problems,  it  is  expedient  to  prepare  the 
values  of  dv  log  p^  d^  log  />2,  a' ',  b1 ',  c',  e,  E,  /,  F,  and  to  arrange 
them  in  tables. 


464 


SOLAR   ECLIPSES. 


For  our  example  of  the  eclipse  of  July  18,  1860,  with  the 
values  of  d  given  on  p.  454,  we  form  the  following  table  by  the 
equations  (495): 


d, 

log/1! 

*i 

los  th 

0* 

21°  T  39".5 

9.9987324 

20°  53'  58".0 

9.9998143 

1 

1  14  .0 

23 

53  32  .6 

45 

2 

0  48  .5 

22 

53   7  .3 

46 

3 

0  22  .9 

21 

52  41  .8 

47 

4 

20  59  57  .4 

20 

52  16  .4 

48 

5 

59  31  .8 

19 

51  50  .9 

50 

The  values  of  x',  y',  and  £',  required  in  (511),  derived  also  from 
the  eclipse  tables  on  p.  454,  by  the  method  of  Art.  75,  are  as 
follows : 


x' 

y' 

P 

0* 

+  0.545277 

—  0.160108 

—  0.000038 

1 

5312 

0486 

061 

2 

''§310 

0846 

084 

3 

5256 

1188 

107 

4 

5134 

1512 

130 

5 

4928 

1818 

154 

Hence,  by  (511)  we  find  the  values  of  a',  bf,  cf  to  be  as  follows. 
The  values  for  interior  contacts  are  seldom  required. 


For  exterior  contacts. 

For  interior  contacts. 

a' 

b' 

cf 

a' 

V 

c' 

0* 

4-  0.001356 

4-  0.050342 

4-  0.631779 

4-  0.001350 

4-  0.050342 

4-  0.631165 

1 

4-  0.000766 

4-  0.101816 

4-  0.616776 

4-  0.000762 

4-  0.101816 

4-  0.616162 

2 

4-  0.000175 

4-  0.153241 

4-  0.601711 

4-  0.000175 

4-  0.153241 

4-  0.601097 

3 

—  0.000415 

4-  0.204612 

4-  0.586571 

—  0.000413 

-f-  0.204612 

4-  0.585957 

4 

—  0.001005 

4-  0.255925 

4-  0.571342 

—  0.001000 

4r  0.255925 

4-  0.570728 

5 

—  0.001595 

4-  0.307171 

4-  0.556010 

—  0.001586 

4-  0.307171 

4-  0.555395 

The  values  of  e9  E,  /,  jP,  for  exterior  contacts,  deduced  from 
these  values  of  bf  and  c',  and  from  d'  =  —  25".5  sin  V,  by  (512), 
are  as  follows: 


OUTLINE   OF   THE    SHADOW. 


465 


E 

log  e 

F 

log/ 

0* 

4°  33'  21" 

9.801939 

—  0°  1'  44" 

9.388244 

1 

9  22  25 

.795965 

tt 

264 

2 

14  17  17 

.793034 

« 

285 

3 

19  13  48 

.793255 

tt 

305 

4 

24   7  46 

.796604 

1    " 

326 

5 

28  55  7 

.802923 

if 

347 

302.  To  illustrate  the  preceding  formulae,  let  us  find  some 
points  of  the  outline  of  the  penumbra  on  the  parth's  surface  at 
the  time  T=  2*  8*  12s.  For  this  time,  we  have 

x  =  —  0.00672  logPl=    9.99873  log  i  =  7.66287 

y  ==  -f  0.57409  dl=  21°    0'  45" 

I  =  -f  0.53673  ^=30   34  13 

Let  us  find  the  points  for  Q  =  50°  and  Q  =  300°.  The  com- 
putation  may  be  arranged  as  follows  : 


Q 

50° 

300° 

By  (502)  : 

a  =  sin  /9  sin  f 

—  0.41788 

-|-  0.45810 

b  =  sin  /?  cos  f 

-f  0.22975 

+  0.30662 

r 

—  61°  11'  52" 

56°  12'  16" 

fi 

28    28  52 

33    27     7 

Hence  by  (504)  : 

£ 

—     5  43 

—     6  59 

ft  -J-  e 

28   23    9 

33    20     8 

By  (505):     logCl  = 

log  COS  03  -|-  e) 

9.94437 

9.92193 

tfCi  sin  § 

+  0.00310 

—  0.00333 

t'd  cos  § 

-f  0.00260 

+  0.00192 

By  (506)  : 

g 

—  0.41478 

+  0.45477 

9i 

+  0.23235 

+  0.30854 

By  (507)  :            log  i 

7j  =  log  c  sin  (7 

9.36614 

9.48931 

log< 

rt  =  log  c  cos  (7 

9.94437 

9.92193 

logc 

9.95901 

•    9.94969 

C 

14°  47'  39" 

20°  16'    9" 

C  -4-  d 

35   48  24 

41    16  54 

By  (508)  :      log  £  = 

log  cos  c>1  sin  # 

7i9.61782 

9.65779 

log  c  cos  (C  -f-  dL)  = 

log  cos  ^  cos  # 

9.86803 

9.82560 

log  tan  ?5» 

7i9.74979 

9.83219 

log  cos  ^j 

9.92764 

9.90803 

log  c  sin  (<?-{-  d1)  = 

log  sin  <pl 

9.72620 

9.76908 

log  tan  y?t 

9.79856 

9.86105 

log  |/(1  —  ee) 

9.99855 

9.99855 

log  tan  </> 

9.80001 

9.86250 

# 

—  29°  20'  20" 

34°  11'  46" 

fJL    ^  W 

59   54  33 

356    22  27 

V 

32    15    3 

36     4  40 

VOL.  I.— 30 


466 


SOLAR   ECLIPSES. 


To  find  whether  the   eclipse   is  beginning  or  ending  at  these 
places,  we  have,  from  the  table  on  p.  465,  for  T=  2h  Sm  12*, 


\oge 

9.7931 

E 

14°  58' 

Q  —  E 

35      2 

285°  2' 

log  e  sin  (Q  —  E~) 

9.5521 

W9.7780 

log/ 

9.3883 

logC/  sin  Q 

9.2170 

7i9.2477 

At  the  first  point,  therefore,  we  have  e  sin  ( Q  —  E)  >  f  t  /  sin  Q, 
and  the  eclipse  is  ending.  At  the  second  point,  we  have 
e  sm(§  — E)  <"J1/sin  §,  and  the  eclipse  is  beginning. 

Rising  and  Setting  Limits. 

303.  To  find  the  rising  and  setting  limits  of  the  eclipse. — By  these 
limits  we  mean  the  curves  upon  which  are  situated  all  those  points 
of  the  earth's  surface  where  the  eclipse  begins  or  ends  with  the 
sun  in  the  horizon.  It  will  be  quite  sufficient  for  all  practical 
purposes  to  determine  these  limits  by  the  condition  that  the 
point  Z  is  in  the  horizon.  This  gives  in  (503)  cos  Z=  0,  or 
f !  =  0,  and,  consequently,  by  (496),  we  have 

£2+7l«=l  (514) 

as  the  condition  which  the  co-ordinates  of  the  required  points 
must  satisfy. 

Now,  let  it  be  required  to  find  the  place  where  this  equation 
is  satisfied  at  a  given  time  T.  Let  x  and  y  be  taken  for  this 
time,  then  we  have,  by  putting  £t=  0  in  (499), 

Z  sin  Q  =  x  —  £ 
I  cos  Q  =  y  —  77 
Let 

m  sin  M  =  x  p  sin  Y  =  £ 

m  cos  M  =  y  p  cos  Y  =  1 

then,  from  the  equations 

I  sin  Q  =  m  sin  M  —  p  sin  Y 
I  cos  Q  =  m  cos  M  —  p  cos  y 

we  deduce,  by  adding  their  squares, 

I*  =  m2  —  2mp  cos  (M  —  y)  +  P9 


J (516) 


2  sin«  t(Jf  -  X   =  1  _  cos 


2mp 


RISING  AND  SETTING  LIMITS.  467 

If  then  we  put  X  =  M  —  f,  we  have 


(517) 


in  which  J^  may  always  be  taken  less  than  90°,  but  the  double 
sign  must  be  used  to  obtain  the  two  points  on  the  surface  of 
the  earth  which  satisfy  the  conditions  at  the  given  time. 

In  this  formula,  m,  M,  and  I  are  accurately  known  for  the 
given  time,  but  p  is  unknown.  It  is  evident,  however,  from 
(514)  and  (515),  that  we  have  nearly  p  =  1,  and  this  value  may 
be  used  in  (517)  for  a  iirst  approximation.  To  obtain  a  more 
correct  value  of  p,  let  us  put  £  =  sin  f\  then,  by  (514),  we  have 
j^—  cos  Y'I  and,  consequently,  since  7=  p^, 

p  sin  Y  =  sin  / 
p  cos  Y  =  pl  cos  Y' 
Hence  we  have 

tan  Y  —  Pi  ^an  Y 


sin  Y        pl  cos 
sin  Y          cos 


(518) 


and  with  this  value  of  p  the  second  computation  of  (517)  will 
give  a  very  exact  value  off.  With  this  second  value  of  f  a  still 
more  correct  value  of  p  could  be  found  ;  but  the  second  approxi- 
mation is  always  sufficient. 

With  the  second  value  of  Y  •>  therefore,  we  find  the  final  value 
of  Y  f  by  the  formula 

tan  /  =  />j  tan  Y 

and  then,  substituting  the  values  £  =  sin  Y  '  ,  Ti\  =  cos  Y  ',  Ci  —  0?  ^n 
(500),  we  have,  for  finding  the  latitude  and  longitude  of  the 
required  points,  the  formulae 

cos  ^jSin  #  =       sin  /  \ 

cos  ^j  cos  i9  =  —  cos  Y'  sin  d^  I 

sin  c?,  =       cos  /  cos  d 


, 

(519) 


tan 

tan      = 


In  the  second  approximation,  we  must  compute  A  and  f  by 
(517)  separately  for  each  place. 


468  SOLAR    ECLIPSES. 

304.  The  sun  is  rising  or  setting  at  the  given  time  at  the 
places  thus  determined,  according  as  &  (which  is  the  hour  angle 
of  the  point  Z)  is  between  180°  and  360°  or  between  0°  and  180°. 

To  determine  whether  the  eclipse  is  beginning  or  ending,  we 
may  have  recourse  to  the  sign  of  P'  (513);  and  it  will  usually  be 
sufficient  for  the  present  problem  to  put  both  af  and  £  =  0  in 
that  expression,  and  then  the  eclipse  is  beginning  or  ending 
according  as  sin  (Q  —  E)  is  negative  or  positive.  Now,  by  (516), 
we  find 


J£)  —  p  sin  (?  -r-  E} 
Hence,  for  points  in  the  rising  or  setting  limits, 

If    m  sin  (M  —  E)  <  p  sin  (j  —  E},  the  eclipse  is  beginning, 
If    m  sin  (M  —  E)  ^>  p  sin  (j  —  E),  the  eclipse  is  ending. 

305.  In  order  to  apply  the  preceding  method  of  determining 
the  rising  and  setting  limits,  it  is  necessary  first  to  find  the 
extreme  times  between  which  the  time  T  is  to  be  assumed,  or 
those  limits  of  T  between  which  the  solution  is  possible.  The 
two  solutions  given  by  (517)  must  reduce  to  a  single  one  when 
the  surface  of  the  cone  of  shadow  has  but  a  single  point  in 
common  with  the  earth's  surface,  —  i.e.  in  the  case  of  tangency  of 
the  cone  and  the  terrestrial  spheroid.  Now,  the  two  solutions 
reduce  to  one  only  wrhen  ^  =  0,  and  both  values  of  f  become  =  M  ; 
but  if  A  =  0,  the  numerator  of  the  value  of  sin  jM  must  also  be 
zero;  and  hence  the  points  of  contact  are  determined  by  the 
conditions 

I  -\-  m  —  p  =  0  and  I  —  m  -f-  p  =  0 

or  by  the  conditions  * 

m  =  p  -j-  I  and  m  =  p  —  I 

There  may  be  four  cases  of  contact,  two  of  exterior  and  two  of 
interior  contact.  The  two  exterior  contacts  are  the  first  and  last, 
or  the  beginning  and  the  end  of  the  eclipse  generally  ;  the  axis  of  the 
shadow  is  then  without  the  earth,  and  therefore  we  must  have 


* 


these  cases  m  =  \/y?  -j-  y2  =  p  ~\-  I. 

The  first  interior  contact  corresponds  to  the  last  point  on  the 
earth's  surface  where  the  eclipse  ends  at  sunrise ;  the  second, 
to  the  first  point  where  it  begins  at  sunset.  But  these  interior 


s  •••  •••• 

B 


RISING   AND    SETTING    LIMITS.  469 

contacts  can  occur  only  when  the  whole  of  the  shadow  on  the 
principal  plane  falls  within  the  earth,  and  for  these  cases,  there- 
fore, we  must  have  m  =  p  —  I. 

For  the  beginning  and  end  generally  we  have,  therefore,  hy 
(515), 

(P  ~\-  0  sin  M  =  x 

(P  -\~  0  cos  M  =  y 

Let  77he  the  time  when  these  conditions  are  satisfied,  and  put 

T=  T0  +  r 

in  which  TQ  is  the  epoch  of  the  eclipse  tables,  for  which  the 
values  of  x  and  y  are  x0  and  ?/0.  Then,  x'  and  yf  being  the  mean 
hourly  changes  of  x  and  y  for  the  time  T,  we  have 

x  =  XQ  -f-  rx' 

y  =  yQ  -f  rtf 

Putting 

m0  sin  Jf0  =  XQ  n  sin  JV7"  =  x' 

m0  cos  Jf0  =  y0  n  cos  N  =  y' 

the  above  conditions  become 

(_p  -j-  0  8^n  M=m0  sin  Jf0  -f-  T  •  w  gin  -^ 
(jP  4~  0  CO8  M  =  m0  cos  Jf0  -}-  r  .  w  cos  ^V" 
whence 

(p  +  1)  sin  (M—N^  —  m0  sin  (  Jf0  —  N) 

(p  _|_  |)  cos  (M  —N)=m0  cos  (  Jf0  —  N)  -f  wr 

so  that,  if  we  put  M  —  N  =  ^,  we  have 

mn  sin  (  Jf  —  JV) 


Bin   4,   r=r 


r  =  £±1 


cos  4  -        cos  (Jf.  - 


in  whicn  cos  ^  may  be  taken  with  either  the  negative  or  the 
positive    sign;    and   it   is  evident  that  the   first  will   give   the 
beginning  and  the  second  the  end  of  the  eclipse  generally. 
For  the  two  interior  contacts  we  have 


sm      = 


m0  sin  (Jf0  —  JV") 

p-l 

P  ~ 

I 

—  cos  4  - 

—  -  cos  (MQ  —  JV) 

n 

470  SOLAR    ECLIPSES. 

These  interior  contacts  cannot  occur  when  p  —  I  is  less  than 
?»0  sin  (M0  —  N),  which  would  give  impossible  values  of  sin  if/. 

In  these  formulae  we  at  first  assume  p  =  1,  and,  after  finding 
an  approximate  value  of  ^,  we  have,  by  (517),  in  which  /I  =  0, 
y  =  Mj  and  in  the  present  problem  M  =  N  -f  ^ :  therefore 

r  =  N+*  (523) 

with  which  p  is  found  by  (518),  and  the  second  computation  of 
(521)  or  (522)  will  then  give  the  required  times.  We  must 
employ  in  (523)  the  two  values  of  $  found  by  taking  cos  ^  with 
the  positive  and  the  negative  sign  ;  and  therefore  different  values 
of  p  will  be  found  for  beginning  and  ending,  so  that  in  the 
second  approximation  separate  computations  will  be  necessary 
f( jr  the  two  cases. 

In  the  first  approximation  the  mean  values  of  x1 ',  #',  and  I 
may  be  used,  or  those  for  the  middle  of  the  eclipse.  With  the 
approximate  values  of  r  thus  found,  the  true  values  of  x',  y' , 
and  I  for  the  time  T=  7"0  +  r  may  be  taken  for  the  second 
approximation. 

After  finding  the  corrected  value  of  ^  we  then  have  also  the 
true  value  of  f  =  N -f-  $  f°r  eacn  point,  and  hence  also  the 
true  value  of  f  by  (518),  with  which  the  latitude  and  longitude 
of  the  points  will  be  computed  by  (519).  For  the  local  apparent 
time  of  the  phenomenon  at  each  place  we  may  take  the  value 
of  d-  in  time,  which  is  very  nearly  the  sun's  hour  angle. 

306.  When  the  interior  contacts  exist,  the  rising  and  setting 
limits  form  two  distinct  enclosed  curves  on  the  earth's  surface. 
If  we  denote  the  times  of  beginning  and  ending  generally,  de- 
termined by  (521),  by  7\  and  T2,  and  the  times  of  interior  con- 
tact, determined  by  (522),  by  T7/  and  7"2',  a  series  of  points  on 
the  rising  limit  will  be  found  by  Art.  303,  for  a  series  of  times 
assumed  between  T^  and  T7/,  and  points  of  the  setting  limit  for 
times  assumed  between  T2f  and  T2. 

When  the  interior  contacts  do  not  exist,  the  rising  and  setting 
limits  meet  and  form  a  single  curve  extending  through  the  whole 
eclipse.  The  form  of  this  curve  may  be  compared  to  that  of  the 
figure  8  much  distorted.  A  series  of  points  upon  it  will  be 
found  by  assuming  times  between  Tl  and  T2. 

307.  EXAMPLE. — Let  us  find  the  rising  and  setting  limits  of 
the  eclipse  of  July  18,  1860. 


RISING    AND    SETTING    LIMITS. 


471 


First. — To  find  the  beginning  and  ending  on  the  earth  gene 
rally,  we  have  for  the  assumed  epoch  TQ  =  2*,  page  455, 


m0  sin  MQ  =  x0  =  —  0.081244 
m0  cos  MQ  =  y0  =  +  0.596075 

which  give 

log  m0  =  9.77930 
M0  =  352°  14'  19" 

log  m0  sin  (MQ  —  N)  =  w9.73938 


m0 
n 


n  sin  N  —  x'  =  -f-  0.5453 
n  cos  N=  y'=  —  0.1608 


log  n  =  9.75474 
J\r=106°  25'.8 

ClL  —  lt}  =  —  0.4336 


For  a  first  approximation,  taking  p  =  1,  we  find,  by  (521), 


p  +  I  =  1.5367 


log  sin  4  =      n9  .5528 


p  +  l 


cos  4,  =  =p   2*.525 


_  2T>  ==  4-   2  .434 


Approx.  beginning   2^=..    23*.909       (July  17) 
"        end  T3  =  _       4  .959       (July  18) 

Taking  cos  ^  negative  for  beginning  and  positive  for  ending, 
we  have  then,  by  (518)  and  (523), 


Beginning. 

End. 

$ 

200°  55'.4 

339°    4'.6 

307    21.2 

85    30.4 

log  tan  7- 

^0.11732 

1.10466 

log  pl 

9.99873 

9.99873 

log  tan  / 

nO.11605 

1.10339 

log  sin  / 

9.89985 

9.99865,5 

log  sin  Y 

990032 

9.99866,3 

log  p 

9.99953 

9.99999 

3  P 

0.99892 

0.99998 

I 

0.53687 

0.53640 

P  +  l 

1.53579 

1.53638 

For  the  above  computed  times  we  further  find 

log  x'  —  log  n  sin  N 

9.73664 

9.73654 

log  ?/'  =  log  n  cos  JV 

n9.20538 

n9.20774 

log  n 

9.75467 

9.75477 

106°  23'  50" 

106°  29'  8" 

472 


SOLAR    ECLIPSES. 


For  a  second  approximation,  therefore,  recomputing  (521),  we 
now  find 


log  sin  4 
log  cos  4 
T 

log  tan  f 

W9.55316 
7i9.97032 
23*.9098 
200°  56'  27" 
307    20  17 
rcO.11629 

W9.55269 
9.97039 
4».9587 
339°    4'  58" 
85    34     6 
1.10942 

and  by  (518) : 

Then,  for  the  latitude  and  longitude  of  the  points,  we  have, 
by  (519), 


tit 

9 

21°  1'42" 
357   9  57 
254  38  57 
102  31  0 
34  38  34 

20°  59'  33" 
72  54  8 
91  35  43 
341  18  25 
4   9  46 

Therefore  the  eclipse  begins  on  the  earth  generally  on  July  17, 
23'1  54W.5  Greenwich  mean  time,  in  west  longitude  102°  31'  0" 
and  latitude  34°  38',  34",  and  ends  July  18,  47t  57W.5  in  longitude 
341°  IS'  25"  and  latitude  4°  9'  46". 

It  is  evident  that  for  practical  purposes  the  first  approximation, 
which  gives  the  times  within  a  few  seconds,  is  quite  sufficient, 
especially  since  the  effect  of  refraction  has  not  yet  been  taken 
into  account.  (See  Art.  327.) 

Secondly. — We  now  pass  to  the  computation  of  the  curve  which 
contains  all  the  points  where  the  eclipse  begins  or  ends  at  sun- 
rise or  sunset.  In  the  present  example,  this  curve  extends 
through  the  whole  eclipse,  since  we  have  mQ  sin  (M0 —  N)  >  1  —  I: 
hence  the  required  points  will  be  found  for  Greenwich  times 
assumed  between  July  17,  23\91  and  July  18,  4\96.  Let  us  take 

the  series 

* 

T,  0*,  0».2,  0».4,  0».6,  0*.8 4».6,  4».8 

The  computation  being  carried  on  for  all  the  points  at  once,  the 
regular  progression  of  the  corresponding  numbers  for  the  suc- 
cessive times  furnishes  at  each  step  a  verification  of  its  correct- 
ness. To  illustrate  the  use  of  the  formula,  I  give  the  computa- 
tion for  T=  2*.0  nearly  in  full.  For  this  time,  we  find,  from 
p.  454  and  p.  464, 


x  =  m  sin  M  =  —  0.08124 
y  =  m  cos  M  —  -j-  0.59608 


I  =  0.53675 


^=21°  0'49" 
\ogpl=  9.99873 


RISING    AND    SETTING   LIMITS. 

and  hence 

M=  352°  14'  21"  log  m  =  9.77931 

Then,  by  (517),  taking  p  =  1,  we  have 


473 


m  =  0.60160 


ar.  co.  log  4mp  9.61863 

I  _|_  m  _  p  =  0.13835 log  9.14098 

I  —  m  4-  p  =  0.93515 log  9.97088 

X  =  26°  49'  log  sin2  }  /I  8.73049 

"With  this  first  approximate  value  of  A  we  find  the  value  of  p  for 
each  of  the  two  points,   by  (518),  as  follows : 


M  ±  A  =  Y 

19°  3' 

325°  25 

log  tan  Y 

9.53820 

9.83849 

log  pl  tan  Y  =  log  tan  / 

9.53693 

9.83722 

,         p,  COS   /         . 

\C\Cf                                \C\Cf   Yl 

9.99887 

9.99914 

cos  Y 

P 

0.99740 

0.99802 

Repeating  (517)  with  these  values  of  p  : 

ar.  co.  log  4  mp 

9.61976 

9.61949 

log  (I  4-  m—  p) 

9.14907 

9.14715 

log  (1  —  m  4-  p) 

9.96967 

9.96996 

log  sin2  J  A 

8.73850 

8.73660 

zh  A 

4-27°    4'    4" 

—  27°    0'  26" 

jf  -4-  ;  —  y 

19    18  25 

325    13  55 

log  tan  Y 

9.54448 

7i9.84148 

log  tan  / 

9.54321 

n9.84021 

Hence,  by  (519), 

* 

135°  45'    4" 

242°  36'  45" 

For  T=  2».  (p.  455),        ^ 

28    31  12 

28    31  12 

//±  —  #  =  w 

252    46     8 

145    54  27 

V 

61    52  35 

50    13  46 

Local  app.  time  =  #  in  time, 

9*     3-.0 

16*  10-45 

Sunset. 

Sunrise. 

To  find  whether  the  eclipse  is  beginning  or  ending  at  these 

points,  we  have,  from  p.  465,  and  by  Art.  304, 

E 

14°  17' 

log  m  sin  (M  —  J5") 

W9.3538 

n9.3538 

log;?  sin    (j  —  JE7) 

8.9406 

n9.8772 

Beginning. 

Ending. 

In  the  same  manner  are  found  the  results  given  in  the  following 
table : 


474 


SOLAR   ECLIPSES. 


SOLAR  ECLIPSE,  July  18,  I860.— RISING  AND  SETTING  LIMITS. 


Greenwich 
Mean  Time. 

Latitude. 
* 

Long.  W.  from 
Greenwich. 

u 

Local 
App.  Time. 

tf 

o*.o 

2 
A 
.6 

.8 

-|-  44°  27' 
52    34 
58      1 
62    10 
65    21 

110°  35' 
121    33 
132    21 
144      2 
157      6 

16*  31-.7 

15   59  .8 
15   28  .7 
14   53  .9 
14   13  .7 

Begins  at  Sunrise. 

it                     tt 
it                    tt 
tt                   <( 
a                   tt 

1.0 
.2 

.4 
.6 

.8 

67    36 
68    49 
68    58 
67    55 
65    37 

171    46 

187    56 
204    56 
221    51 
237    54 

13   27  .0 
12   34  .4 
11   38  .3 
10  42  .7 
9   50  .5 

a                    it 
it                   tt 

"           Sunset. 

tt               tt 

tt               tt 

2.0 
.2 
.4 
.6 

.8 

61    53 
56    16 
48      5 
37    15 
25      6 

252    46 
266    33 
279    17 

290    36 
300    12 

9     3  .0 

8   19  .9 
7   41  .0 

7     7  .7 
6   41  .3 

tt               it 
it               tt 
tt               tt 
tt               tt 
n               tt 

3  .0 
.2 
.4 
.6 

.8 

13    36 
+    3    59 
—    3    24 
—    8    43 
—  12    14 

308    12 
315      0 
320    50 
325    53 
330    17 

6   21  .3 
6     6  .1 
5   54  .8 
5  46  .5 
5  41  .0 

tt               it 
ti               tt 
tt               it 
tt               it 
tt               tt 

4.0 
.2 
.4 

.6 

.8 

—  14    11 
—  14    48 
—  14      6 
—  11    56 
—    7    32 

334      4 
337    19 
340      2 
342      9 
343    25 

5  37  .8 
5   36  .8 
5   38  .0 
5   41  .5 

5   48  .4 

it               tt 
it               tt 

Ends            " 

tt               tt 

a                   (t 

0  .0 
.2 

.4 
.6 

.8 

-|-  25    45 
20      1 
17    16 
16      7 
16    17 

99    10 
99    33 
101    22 

103    52 

106    56 

17    17  .4 
17   27  .9 
17    32  .6 
17    34  .6 
17   34  .3 

Begins  at  Sunrise. 

tt                     tt 
tt                  tt 

Ends           " 

tt               u 

1  .0 
.2 
.4 

.6 
.8 

17    46 
20    42 
25    17 
31    45 
40      0 

110    34 
114    50 
119    57 
126    14 
134    15 

17   31  .8 
17   26  .7 
17   18  .3 
17     5  .2 
16   45  .0 

tt               tt 
tt                tt 
tt               it 
tt               tt 
tt               n 

2.0 
.2 
.4 

.6 

.8 

50    14 

60    21 
67    27 
68    55 

66    27 

145    54 
163    47 
191    43 
224    18 
249      7 

16   10  .5 
15  -10  .9 
13   31  .2 
11  32  .9 
10     5  .6 

u               u 
it               it 
tt               tt 

"           Sunset. 

tt               tt 

' 


CURVE    OF    MAXIMUM 


THE    HORIZON. 


475 


SOLAR  ECLIPSE,  July  18,  I860.— RISING  AND  SETTING  LIMITS.— (Continued.) 


Greenwich 
Mean  Time. 

Latitude. 
t 

Long.  W.  from 
Greenwich. 

u 

Local 
App.  Time. 

tf 

3A.0 
2 

A 
.6 

.8 

-f  62°  43' 
58    44 
54    42 
50    35 
46    21 

265°  37' 
277    27 
286    49 
294    47 
301    53 

9»  11*.  6 
8    36   .3 

8    10  .8 
7   51  .0 
7   34  .6 

Ends  at  Sunset. 

a 

a 
a 
tt 

4  .0 
2 
.4 
.6 

.8 

41    55 

37    10 
31    57 
25    55 
18    11 

308    26 
314    40 
320    43 
326    48 
333    18 

7   20  .3 
7     7  .4 
6   55  .2 
6  42  .9 

6   28  .9 

n 
u 

tt                  u 
11                  tt 
tt                     « 

These  points  being  projected  upon  a  chart  (see  p.  504),  the 
whole  curve  may  be  accurately  traced  through  them.  It  will  be 
seen  that  the  method  of  -assuming  a  series  of  equidistant  times 
gives  more  points  in  those  portions  of  the  curve  where  the 
curvature  is  greatest  than  in  other  portions,  thus  facilitating  the 
accurate  delineation  of  the  curve.  This  advantage  appears  to 
have  been  overlooked  by  those  who  have  preferred  methods 
(such,  for  example,  as  HANSEX'S)  in  which  a  series  of  equidistant 
latitudes  is  assumed. 

308.  The  preceding  computations  have   been   made  for  the 
penumbra  ;  but  we  may  employ  the  same  method  to  determine 
the   rising   and   setting   limits   of  total   or   annular  eclipse  by 
employing  in  the  formulae  the  value  of  I  for  interior  contacts. 
These  limits,  however,  embrace  so  small  a  portion  of  the  earth's 
surface  that  they  are  practically  of  little  interest. 

Curve  of  Maximum  in  the  Horizon. 

309.  To  find  the  curve  on  which  the  maximum  of  the  eclipse  is  seen 
at  sunrise  or  sunset.  —  When  a  point  of  the  earth's  surface  whose 
co-ordinates  are  f,  37,  and  £  is  not  on  the  surface  of  the  cone  of 
shadow,  but  at  a  distance  J  from  the  axis  of  the  cone,  we  have 
the  conditions  (485), 


cos  Q  =  y  — 


(524) 
*-       > 


476  SOLAR    ECLIPSES. 

The  amount  of  obscuration  depends  upon  the  distance  by 
which  the  place  is  immersed  within  the  shadow,  that  is,  upon  the 
distance  L  —  J,  L  being  the  radius  of  the  shadow  on  the  parallel 
plane  at  the  distance  £  from  the  principal  plane.  For  the 
maximum  of  the  eclipse,  therefore,  we  have  the  condition 


dT         dT 

Differentiating  the  above  equations  relatively  to  the  time,  and 
denoting  the  derivatives  of  x,  y,  &c.  by  accents,  as  in  Art.  300, 
we  have 


Bn     _j  cos 


which  give 

— —  —  (#' —  £')  sin  Q  -f-  (if —  >/)  cos  § 

The  equation  L=l  —  if  gives 

—  --r— i" 

JT  = 
and,  therefore, 

T—  *:'  —  (xr—  £')  sin  (2  —  (/—  V)  cos  §  =  0          (525) 

or,  by  (510), 

P'=Q  (526) 

This  is,  therefore,  the  general  condition  which  characterizes 
the  maximum  of  the  eclipse  at  a  given  time.  In  the  present 
problem  we  have  also  the  condition  that  the  sun  is  in  the  horizon, 
for  which  we  may,  as  in  Art.  303,  substitute  the  condition  f  L  =  0. 
Since,  however,  the  instant  of  greatest  obscuration  is  not  subject 
to  any  nice  observation,  a  very  precise  solution  of  the  problem 
is  quite  unimportant,  and  we  may  be  satisfied  with  the  approxi- 
mate solution  obtained  by  supposing  f  =  0,  and  at  the  same 
time  neglecting  the  small  quantity  a'  in  Pr.  The  condition 
(526)  will  then  be  satisfied  when  in  (513)  we  have 

sin  (Q  —  E)  =  0 
that  is,  when 

Q  =  E  or  Q  =  180°  -f  E 


CURVE   OP    MAXIMUM    IN   THE    HORIZON.  477 

Hence,  for  any  given  time,  the  conditions  (524)  become 


±  A  sin  E  =  x  — 
±  J  cos  E  =  y  — 


sin    *  = 


which  with  the  condition 


must  determine  the  required  points  of  our  curve.  The  angle  E 
is  here  known  for  the  given  time,  being  directly  obtained  from 
its  tabulated  values,  but  A  is  unknown.  Putting,  as  in  the 
preceding  problem, 

m  sin  M  =  x  p  sin  Y  —  £ 

m  cos  M  =  y  p  cos  Y  =  y 

we  have 

dt  J  sin  E  =  m  sin  M  —  p  sin  Y 

±   J  COS  ^  =  771  COS  M  —  p  COS  Y 

whence 

0  =  m  sin  (M  —  E)  —  p  sin  (y  —  J£) 

±  J  =  m  cos  (Jlf  —  _£*)  —  p  cos  (y  —  E) 

Therefore,  putting  ^  =  Y  —  E,  we  have 

m  sin  ( Jf  —  .Z7) 

(527) 

±  J  =  m  cos  (Jf  —  J£)  —  ^?  cos 

The  first  of  these  equations  will  give  two  values  of  <\J/,  since  we 
may  take  cos  ij/  with  the  positive  or  the  negative  sign  ;  but,  as 
only  those  places  satisfy  the  problem  which  are  actually  within 
the  shadow,  we  must  have  A  <  I,  or,  at  least,  A  not  greater  than  I. 
That  value  of  $  which  would  give  J>  I  must,  therefore,  be 
excluded :  so  that  in  general  we  shall  have  at  a  given  time  but 
one  solution. 

It  will  be  quite  accurate  enough,  considering  the  degree  of 
precision  above  assigned,  to  employ  in  (527)  a  mean  value  of  p, 
or,  since  p  falls  between  pl  and  unity,  to  take  log  p  =  Jlog  pr 
But,  if  we  wish  a  more  correct  value,  we  have  only  to  take 

Y  =  4  +  E  (528) 

and  then  find  p  as  in  (518) ;  after  which  (527)  must  be  recom- 
puted. 


478  SOLAR   ECLIPSES.         f 

Having  found  the  true  value  of  $  by  (527),  and  of  7-  by  (528), 
we  then  have  f  by  the  equation 

tan  Y'  =  Pi  tan  T 

and  the  latitude  and  longitude  of  each  point  of  the  curve  by  (519). 
The  limiting  times  between  which  the  solution  is  possible  will 
be  known  from  the  computation  of  the  rising  and  setting  limits, 
in  which  we  have  already  employed  the  quantity  m  sin  (M  —  E)\ 
and  the  present  curve  will  be  computed  only  for  those  times  for 
which  m  sin  (M  —  JS)  <  I.  •  These  limiting  times  are  also  the  same 
as  those  for  the  northern  and  southern  limiting  curves,  which 
will  be  determined  in  Art.  313. 

310.  The  degree  of  obscuration  is  usually  expressed  by  the 
fraction  of  the  sun's  apparent  diameter  which  is  covered  by  the 
moon's  disc.  When  the  place  is  so  far  immersed  in  the  penumbra 
as  to  be  on  the  edge  of  the  total  shadow,  the  obscuration  is  total  ; 
in  this  case  the  distance  of  the  place  from  the  edge  of  the 
penumbra  is  equal  to  the  absolute  difference  of  the  radii  of  the 
penumbra  and  the  umbra,  that  is,  to  the  algebraic  sum  L  +  L^ 
Z/L  denoting  the  radius  of  the  umbra  (which  is,  by  Art.  2'93, 
negative)  ;  but  in  any  other  case  the  distance  of  the  place  within 
the  penumbra  is  L  —  4:  hence,  if  D  denotes  the  degree  of 
obscuration  expressed  as  a  fraction  of  the  sun's  apparent 
diameter,  we  shall  have,  very  nearly, 

,-L  .(529) 


This  formula  may  also  be  used  when  the  eclipse  is  annular,  in 
which  case  Lt  is  essentially  positive  ;  and  even  when  A  is  zero, 
and  the  eodipse  consequently  central,  the  value  of  D  given  by 
the  formula  will  be  less  than  unity,  as  it  should  be,  since  in  that 
case  there  is  no  total  obscuration. 
In  the  present  problem  we  have 

D  =  l~A  (529*) 

*+*i 

in  which  I  and  ^  are  the  radii  of  the  penumbra  and  umbra  on 
the  principal  plane,  as  found  by  (488). 

EXAMPLE.—  In  the  eclipse  of  July  18,  1860,  compute  the  curve 
on  which  the  maximum  of  the  eclipse  is  seen  in  the  horizon. 


CURVE   OF   MAXIMUM   IN   THE   HORIZON. 


479 


In  the  computation  of  the  rising  and  setting  limits,  the 
quantity  m  sin  (M—  E)  was  less  than  unity  only  from  T=  0\6 
to  T==  4\2 :  so  that  the  present  curve  may  be  computed  for  the 

series  of  times  0\6,  0\8 4\0,  4\2.  For  an  approximate 

computation  we  may  take  log  p  =  Jlog  pl=  9.9994,  and  employ 
only  four  decimal  places  in  the  logarithms  throughout. 

The  computation  for  T  =  2*  is  as  follows.  For  this  time  we 
have  already  found  (p.  473) 

log  m 
M 
E 

Hence,  by  (527),  M  —  E 

log  m  sin  (Jtf  —  E) 

\ogp 

log  sin  4 

log  cos  ^ 

log  p  cos  ^ 

log  m  cos  (M  —  E) 

m  tjos  (Jf  —  E} 
p  cos  4 

A 


9.7793 
352°  14'.4 
14    17.3 
337    57.1 
n9.3538 
9.9994 


?i9.3544 
9.9886 
9.9880 
9.7463 


+  0.5575 
4-  0.9727 


0.4152 


Here,  if  cos  if>  were  taken  with  the  negative  sign  we  should 
find  4=  1.5302,  which  is  greater  than  I.  Taking  it,  therefore, 
with  the  positive  sign  only,  we  have 


log  Pl  =  9.9987 
with  which  we  find,  by  (519), 


log  tan  Y 
log  tan  / 


App.  time  =  $  in  time 


—  13°    4'.3 
+    1    13. 
8.3271 

8.3258 


176°  37'.2 
28    31.2 

211    54 
69      1 

11*  46-5 
Sunset. 


To  express  the  degree  of  obscuration  according  to  (529*)  we 
have,  taking  the  mean  values  of  I  and  ^  (p.  454), 


I  ==  0.5366 
*!  =  —  0.0092 
L  =  0.5274 


—  J  ==  0.1214 

_  0.1214 

~  0.5274 


=  0.23 


In  the  same  manner  all  the  following  results  are  obtained : 


480 


SOLAR   ECLIPSES. 


SOLAR  ECLIPSE,  July  18,  I860.— CURVE  OF  MAXIMUM  OF  THE  ECLIPSE 

IN  THE  HORIZON. 


Greenwich 
Mean  T. 

Latitude. 

$ 

Long.  W.  from 
Greenwich. 

u 

App.  Local 
Time. 

tf 

Degree  of 
Obscuration. 

D 

0\6 
0  .8 
1  .0 
1  .2 
1  .4 

+  24°  44' 
37   47 
47     3 
54   31 

60   38 

107°  41' 
117    47 
127    49 
139      1 
152    24 

17*  19-.8 
16    50  .9 
16    22  .8 
15    50  .0 
15     8  .5 

0.30 
.76 
.97 
.74 
.56 

1  .6 
1  .8 
2.0 
2  .2 
2  .4 

65   20 
68    16 
69     1 
67    34 
64   20 

169      0 
189    16 
211    54 
233    32 
251    42 

14   14  .1 

13      5  .0 
11   46  .5 
10   31  .9 
9   31  .3 

.41 
.31 
.23 
.18 
.17 

2  .6 

2  .8 
3.0 
3  .2 
3  .4 

59    55 
54   41 
48   52 
42   35 
35   49 

266    11 
277    50 

287    31 
295    56 
303    30 

8   45  .3 
8    10  .8 
7   44  .0 
7    22  .4 
7     4  .1 

.17 
.21 

.28 
.37 
.50 

3  .6 
3  .8 
4  .0 
4  .2 

28    28 
20    21 
+  11     2 
—    0   45 

310    33 
317    22 
324    15 
331    14 

6   47  .9 
6   32  .6 
6   17  .2 
6     1  .1 

.67 

.89 

.87 
.48 

Northern  and  Southern  Limiting  Curves. 

311.  To  find  the  northern  and  southern  limits  of  the  eclipse  on  the 
earth's  surface. — These  limits  are  the  curves  in  which  are  situated 
all  the  points  of  the  surface  of  the  earth  from  which  only  a  single 
contact  of  the  discs  of  the  sun  and  moon  can  be  observed,  the 
moon  appearing  to  pass  either  wholly  south  or  wholly  north  of 
the  sun.  They  may  also  be  defined  as  curves  to  which  the  out- 
line of  the  shadow  is  at  all  times  in  contact  during  its  progress 
across  the  earth. 

The  solution  of  this  problem  is  derived  from  the  consideration 
that  the  simple  contact  is  here  the  maximum  of  the  eclipse,  so 
that  we  must  have,  as  in  (526), 


and  consequently,  by  (513), 
a'  -f  e  sin  (Q  - 


(530) 


NORTHERN  AND  SOUTHERN  LIMITS.          .  481 

For  any  given  time  7",  therefore,  we  are  to  find  that  point  of 
the  outline  of  the  shadow  on  the  surface  of  the  earth  for  which 
the  value  of  Q  and  its  corresponding  f  satisfy  this  equation. 
This  can  be  effected  only  indirectly,  or  by  successive  approxima- 
tions. For  this  purpose,  we  must  know  at  the  outset  an  approxi- 
mate value  of  Q;  and  therefore,  before  proceeding  any  further, 
we  must  show  how  such  an  approximate  value  may  be  found. 

We  can  readily  determine  sufficiently  narrow  limits  between 
which  Q  may  be  assumed.  For  this  purpose,  neglecting  a'  in 
(530),  as  well  as  Fy  which  are  always  very  small,  we  have, 
approximately, 


The  extreme  values  of  f  are  £  =  0  and  £  =  1.     The  first  gives 
sin  (Q  —  E}  —  0,  and  therefore  for  a  first  limit  we  have 

Q  =  E  or  Q  =  180°  -f-  ^ 

The  second  gives 

e  sin  (Q  —  E)  —/sin  Q 
whence 

tan  (Q  —  £  .#)  =  -  tZ  tan  J  J0 
Put 

tan  4  =  —        tan  J  E 

then  the  equation  tan  (Q  —  %E]  =  tan  -J/  gives  for  our  second 
limits 

Q  —  ±E-\-%  or  Q  =  180°  -f.  %E -{-  4 

To  compute  <\J/  readily,  put 


tan  v  = 
e 


then 


tan  4  =  tan  (45°  -f  v)  tan  \E 
and  Q  is  to  be  assumed 

between  E  and  £  E  -f-  4 
or  between  180°  -f  E  and  180°  -f  £  E  -\- 
VOL.  L— 31 


(531) 


482  SOLAR   ECLIPSES. 

These  limits  may  be  computed  in  advance  for  the  principal 
hours  of  the  eclipse  from  the  previously  tabulated  values  of 
E,  e,  and  /,  and  an  approximate  value  of  Q  may  then  be  easily 
inferred  for  a  given  time  with  sufficient  precision  for  a  first 
approximation. 

When  the  shadow  passes  wholly  within  the  earth,  there  are 
two  limiting  curves,  northern  and  southern.  For  one  of  these 
Q  is  to  be  taken  between  E  and  f  E  -\-  ^ ;  for  the  other,  between 
180°  -f  E  and  180°  -f  J  E  -f  ^.  Since  E  is  always  an  acute  angle, 
positive  or  negative,  it  follows  that  when  Q  is  taken  between 
J£and  J  E  -\-  ij/,  its  cosine  is  in  general  positive,  while  it  is  nega- 
tive in  the  other  case.  The  equation  37  =  y  —  (I  —  /£)  cos  Q 
shows  that  rt  will  be  less  in  the  first  case  and  greater  in  the 
second,  and  hence  the  values  of  Q  between  E  and  J  E  +  ^  belong 
to  the  southern  limit,  and  the  values  of  Q  between  180°  +  E  and 
180°  +  i  E  +  ^  belong  to  the  northern  limit. 

There  is  only  one  limit,  northern  or  southern,  when  one  of  the 
series  of  values  of  Q  would  give  impossible  values  of  £  in  the 
computation  of  the  outline  of  the  shadow  by  Art.  298.  But  when 
the  rising  and  setting  limits  have  been  determined,  the  question 
of  the  existence  of  one  or  both  of  the  northern  and  southern 
limits  is  already  settled  ;  for  if  the  rising  and  setting  limits  extend 
through  the  whole  eclipse  in  north  latitude,  only  the  southern 
limiting  curve  of  our  present  problem  exists,  and  vice  versa; 
while  if  the  rising  and  setting  limits  form  two  distinct  curves, 
we  have  both  a  northern  and  southern  limiting  curve ;  and  the 
latter  must  evidently  connect  the  extreme  northern  and  southern 
points  respectively  of  the  two  enclosed  rising  and  setting  curves. 
In  our  example  of  the  eclipse  of  July  18,  1860,  there  exists  only 
the  southern  limiting  curve  of  the  present  problemvthe  penum- 
bral  shadow  passing  over  and  beyond  the  north  pole  of  the  earth. 

Having  assumed  a  value  of  §,  we  find  £L  by  the  equations  (502), 
(504)  and  (505),  and  then  £  by  (509).  This  computed  value  of  £ 
and  the  assumed  value  of  Q  being  substituted  in  (530),  this  equa- 
tion will  be  satisfied  only  when  the  true  value  of  Q  has  been 
assumed.  To  find  the  correction  of  Q,  let  us  suppose  that  when 
the  equation  has  been  computed  logarithmically  we  find 

log  C/sin  (Q  —  F)  —  log  [V  +  e  sin  (Q  -  Eft  =x 
If  then  dQ  and  d£  are  the  corrections  which  Q  and  f  require  in 


NORTHERN   AND    SOUTHERN    LIMITS.  483 

order  to  reduce  x  to  zero,  we  have,  by  differentiating  this  equation, 


a'  -\-  e  sin  (Q  —  E}      A 

in  which  A  is  the  reciprocal  of  the  modulus  of  common  logarithms. 
In  this  differential  equation  we  may  neglect  af  without  sensibly 
affecting  the  rate  of  approximation.     If  then  we  put 

d 
we  shall  have 


dQ  = 


cot  (Q  ~  E)  —  cot  (Q  —  F)  -f  g 


This  value  of  dQ  is  yet  to  be  reduced  to  seconds  by  multiplying 
it  by  cosec  V  or  206265". 

To  find  g,  we  may  take,  as  a  sufficiently  exact  expression  for 
computing  dQy 

~^dQ 

and  by  differentiating  (502)  (omitting  the  factor  plt  which  will 
not  sensibly  affect  #), 

cos  p  sin  Y  dp  -\-  sin  p  cos  Y  dy  =  —  I  cos  Q  dQ 
cos  p  cos  Y  dt3  —  sin  p  sin  Y  dy  =       I  sin  Q  dQ 

whence,  by  eliminating  <fy, 


dQ  cos/9 

By  (505)  a  sufficiently  exact  value  of  ^  for  our  present  pur- 
pose is 

C,  =  cos  p 
whence 

^  =  -sin^ 
dQ  dQ 

g  =  I  sin  p  sec'  /9  Bin  (§  —  r)  (532) 

Putting,  finally, 

a  =  cot  (§  —  J0)—  cot  (C  —  .P)  =  -  sm  (^  —  ^)  - 


484  SOLAR    ECLIPSES. 

we  liave 


&  +  9 

in  which  5.67664  is  the  logarithm  ofAX  206265". 

When  the  true  value  of  Q  has  thus  been  found,  the  corre- 
sponding latitude  and  longitude  on  the  earth's  surface  are  found 
as  in  Art.  298. 

312.  The  preceding  solution  of  this  problem  (which  is  com- 
monly regarded  as  one  of  the  most  intricate  problems  in  the 
theory  of  eclipses)  is  very  precise,  and  the  successive  approxi- 
mations converge  rapidly  to  the  final  result.  For  practical  pur- 
poses, however,  an  extremely  precise  determination  of  the  limit- 
ing curves  of  the  penumbra  is  of  little  importance,  since  no 
valuable  observations  are  made  near  these  limits.  I  shall,  there- 
fore, now  show  how  the  process  may  be  abridged  without  making 

any  important  sacrifice  of  accuracy. 
In  the  first  place,  it  is  to  be  observed 
that  great  precision  in  the  angle  Q 
is  unnecessary.  If  LM,  Fig.  43,  is 
the  limiting  curve  which  is  tangent 
at  A  to  the  shadow  whose  axis  is  at 
(7,  and  if  Q  is  in  error  by  the  quan- 
tity ACA',  the  point  determined  will  be  (nearly)  A'  instead  of  A. 
Now,  although  A'  may  be  at  some  distance  from  A,  it  is  evident 
that  it  will  still  be  at  a  proportionally  small  distance  from  the 
limiting  curve.  In  fact,  we  may  admit  an  error  of  several 
minutes  in  the  value  of  Q  without  sensibly  removing  the  computed 
point  from  the  curve.  The  equation  (530),  which  determines  §, 
may,  therefore,  without  practical  error  be  written  under  the 
approximate  form 

esin(<2  —  J£)=  :t/sin  Q 

and  in  this  we  may  employ  for  £t  the  value 

C,  =  cos  ft 

Hence,  having  found  ft  from  (502)  by  employing  the  first  assumed 
value  of  §,  we  then  have 

sin  (Q—E)  __/cosff 
sin  Q  e 


NORTHERN  AND  SOUTHERN  LIMITS.  485 

whence 


e  —  /  cos  ft 

by  which  a  second  and  more  correct  value  of  Q  can  be  found. 
This  equation  will  be  readily  computed  under  the  following  form  : 


tan  (Q  —  iJS)  =  tan  (45°  +  J)  tan  IE 


The  value  of  Q  thus  determined  may  be  regarded  as  final,  and 
we  may  then  proceed  to  compute  the  latitude  and  longitude  by 
the  equations  (502)  to  (508).  In  this  approximate  method,  loga- 
rithms of  four  decimal  places  will  be  found  quite  sufficient. 

313.  For  the  computation  of  a  series  of  points  by  the  preceding 
method,  it  is  necessary  first  to  determine  the  extreme  times 
between  which  the  solution  is  possible.  It  is  evident  that  the 
first  and  last  points  of  the  curve  are  those  for  which  f  t  =  0,  and, 
consequently,  Q  =  E,  or  Q  =  180°  -j-  E.  It  is  easily  seen  that 
these  points  are  also  the  first  and  last  points  of  the  curve  of 
maximum  in1  the  horizon  (Art.  309),  and,  therefore,  the  limiting 
times  are  here  the  same  as  for  that  curve.  If,  however,  we  wish 
to  determine  these  limiting  times  independently  (that  is,  when 
the  rising  and  setting  limits  have  not  been  previously  computed), 
the  following  approximative  process  will  give  them  with  all  the 
precision  necessary. 

Since  Q  =  E,  or  =  180°  -f-  E,  we  have,  at  the  required  time, 

e  =  x  +  lsinE  1 

r)  =  y  =F  I  cos  E  j 

together  with  the  condition  (514),  for  which  we  may  here  employ 

?  +  ^  =  1 

If  we  put  £  =  sin  f,  this  condition  gives  y  =  cos  f.  We  have, 
by  (512), 

sin  E  —  —  cos  E  =  — 

e  e 

and  we  may  here  regard  e  as  constant.  Let  the  required  time 
be  denoted  by  T=  TQ  -f  r,  T0  being  an  assumed  time  near  the 
middle  of  the  eclipse.  Let  60',  <?0',  be  the  values  of  bf  and  c'  for 


486  SOLAR    ECLIPSES. 

the  time  Tw  and  denote  their  hourly  changes  by  b"  and  c" ;  then 
we  have,  for  the  time  T, 


and  hence,  JE0  being  the  tabulated  value  of  E  for  the  time  Tw 

b"                                                       c" 
sin  E  =  sin  E0  -j r  cos  E  =  cos  EQ  -j r 

6  6 

If,  also,  #0,  y0,  are  the  values  of  x  and  y  for  the  time  Tw  xr  and  j/' 
their  hourly  changes,  we  have 

x  =  xti  -f  x'  T  y  =  y0  -j-  y'  r 

and  the  equations  (536)  become 

sin  Y  =  XQ+  I  sin  EQ  -j-  I  -r'  T  —  &" 

\  e 


cos  r 


=  y0  q=  /  cos  J5;o  -f  I  yf  rp  _  c"  J 


Let  m,  Jf,  7i,  ^V,  be  determined  by  the  equations 
m  sin  M  =  XQ  ^  I  sin  E0 
m  cos  M  =  3/0  =F  Z  cos  JS',, 

fflsin^=a;--l6,,  • 

.  0 

=  /^c" 

e 


(537) 


in  which  the  upper  sign  is  to  be  used  for  the  southern  and  the 
lower  sign  for  the  northern  limit  ;  then,  from  the  equations 

sin  r  =  m  sin  M  -\-  n  sin  JV.  r 
cos  f  =  m  cos  M  -j-  n  cos  N  .  T 
we  derive 


sn    Y  —  -       =  m  8 

cos  (Y  —  N)  =.  m  cos  (M  —  J?r)  -j-  nr 

Hence,  putting  f  —  N=  ^, 

sin  4  ==  m  sin  (M  —  N) 

—  N) 


It  is  evident  that  cos  ^  is  to  be  taken  with  the  negative  sign  for 
the  first  point  and  with  the  positive  sign  for  the  last  point  of 
the  curve. 


NORTHERN  AND  SOUTHERN  LIMITS. 


487 


To  find  the  latitude  and  longitude  of  the  extreme  points,  we 
take  f  =  N  -\-  i^,  tan  f  =  pl  tan  f,  and  proceed  by  (519). 

EXAMPLE.  —  To  find  the  southern  limit  of  the  eclipse  of  Jul} 
18,  1860. 

First.  To  find  the  extreme  times.  —  Taking  TQ  =  2*,  we  have, 
from  our  tables,  pp.  454,  455,  and  pp.  464,  465, 


XQ  =  _  0.0812 


yo  =  4-  0.5961 
I  =       0.5367 
EQ=       14°  17' 
log  e  =       9.7977 


x'  =  4-  0.5452 
y'  =  —  0.1610 

6"  =  +  0.0514 
c"  =  —  0.0151 


where  we  take  mean  values  of  x',  yf  ,  &c.     From  these  we  find 
by  (537),  taking  the  upper  signs  in  the  formulae, 


log  m  =  9.3555 
log  n  =  9.7182 

Hence,  by  (538), 

log  sin  (M  —  JV)  =  ?i8.7354 
log  sin  4,  =  w8.0909 
log  cos  4  =  0.0000 


M  =289°  35' 
JV=106    28 
M—  ^=183      7 

log  cos  ( M  —  N)  —  n9.9994 
_mco8(Jf-JT)  = 

M 

^A^T  1.013 


r  =  —  1  .480 
or  T  =  4-  2  .346 

Therefore,  for  the  first  and  last  points  of  the  curve  we  have, 
respectively,  the  times 

T±  =  2*  —  1\480  =  0*.520 
Ta  =  2    4-2  .346  =  4  .346 

To  find  the  latitude  and  longitude  of  the  extreme  points  corre- 
sponding to  these  times,  we  have 


r  =  R  +  4 

log  tan  Y 
log  /?,  =  9.9987        log  tan  / 


ft 


First  Point. 

Last  Point. 

180°  42' 

—       0°  42' 

287    10 

105     46 

nO.5102 

nO.5492 

wO.5089 

wO.5479 

21°    T.4 

20°  59'.8 

6    19.2 

63    42.7 

488 

Hence,  by  (519), 


SOLAR    ECLIPSES. 


102°  40' 
16      5 


339°  30' 
—    14    47 


Second.  To  find  a  series  of  points  on  the  curve. — We  begin  by 
computing  the  limits  of  Q  for  the  hours  0*,  1*,  2*,  3*,  4*,  5*.  Thus, 
for  0A  we  have,  from  the  table  p.  465,  and  by  (531), 


T 

C* 

log/ 

9.3882 

loge 

9.8019 

log  tan  v 

9.5863 

V 

21°    5'.6 

JhB 

2    16.7 

log 

tan  (45°  +  v) 

0.3533 

log  tan  j  12 

8.5997 

log  tan  4 

8.9530 

4 

5°    7'.7 

J2£+  4- 

7    24.4 

For  the  southern  limiting  curve,  Q  falls  between  .Z?and  %E-\-  4/, 
i.e.,  for  0A,  between  4°  33'  and  7°  24'.  In  the  same  manner  we 
form  the  other  numbers  of  the  following  table : 


T 

Lower  limit  of  Q. 

Upper  limit  of  Q. 

0* 

4°  33' 

7°  24' 

1    • 

9    22 

15    18 

2 

14    17 

23    13 

3 

19    14 

30    53 

4 

24      8 

38      4 

5 

28    55 

44    36 

The  points  of  the  curve  are  to  be  computed  for  times  between 
0\520   and   4\346,  and  we   shall,  therefore,  assume  for  T  the 

series  0*.6,  0*.8,  1*.0 4*.0,  4*.2,  which,  with  the  extreme 

points  above  computed,  will  embrace  the  whole  curve. 

Instead  of  determining  Q  for  each  of  these  times  by  the 
method  of  Art.  312,  it  will  be  sufficient  to  determine  it  for  the 
hours  l/l,  2*,  3A,  4*,  and,  hence,  to  infer  its  values  for  the  inter- 
vening times.  Thus,  for  T=  1*,  assuming  Q=  12°,  which  is  a 


NORTHERN  AND  SOUTHERN  LIMITS. 


489 


mean  between  its  two  limiting  values,  we  proceed  by  the  equa- 
tions (502),  for  which  we  can  here  use 


sin  ft  sin  f  =  x  —  I  sin  Q 
sin  ft  cosf  =  y  —  I  cos  Q 


as  follows : 


For  TQ=lh. 


Assume  Q 
a  =  x  —  I  sin  Q 
b  =  y  —  I  cos  Q 
log  a  =  log  sin  ft  sin 
log  b  =  log  sin  ft  cos 
log  sin 

We  thus  find, 

for  T=      1» 
Q  =  11°  55', 


X 

—  0.6266 

log  cos  ft 

9.7396 

y 

-j-  0.9170 

log^ 

9.5923 

i 

0.5368 

6 

Q 

12° 

log  tan  v' 

9.3319 

Q 

-  0.7382 

v' 

12°    7M 

Q 

+  0.3920 

|iS 

4    41.2 

r 

W9.8682 

logtan(45°+/) 

0.1894 

r 

9.5933 

tan  \E 

8.9137 

[i 

9.9221 

tan(§—  \E) 

9.1031 

Q  —  \E 

7°  13'.5 

Q 

11    54.7 

2* 
22°  20', 


3* 
30°  16', 


4* 
32°  17'. 


From  these  numbers  we  obtain  by  simple  interpolation  suffi- 
ciently exact  values  of  Q  for  our  whole  series  of  points.  And 
since  it  is  plain  from  Art.  312,  that  even  an  error  of  half  a 
degree  in  Q  will  not  remove  the  computed  point  from  the  true 
curve  by  any  important  amount,  we  may  be  content  to  employ 
the  following  series  of  values  as  final : 


T 

Q 

T 

Q 

T 

Q 

T 

Q 

0*.6 

8° 

1*.6 

18° 

2*.6 

28° 

3*.6 

31° 

0  .8 

10 

1  .8 

20 

2  .8 

29 

3.8 

32 

1  .0 

12 

2.0 

22 

3  .0 

30 

4  .0 

32 

1  .2 

14 

2  .2 

24 

3.2 

30 

4  .2 

32  .5 

1  .4 

16 

2.4 

26 

3  .4 

31 

For  each  time  jTwe  now  take  x,  y,  and  I,  from  the  tables  of 
the  eclipse,  arid,  with  the  value  of  Q  for  the  same  time,  deter- 
mine the  required  point  on  the  outline  of  the  shadow  by  the 


490 


SOLAR    ECLIPSES. 


complete  equations  (502)  to  (508)  inclusive,  the  use  of  which  has 
already  been  exemplified  in  Art,  302.  Employing  only  four 
decimal  places  in  the  logarithms,  we  shall  find  that  the  curve 
may  be  traced  through  the  points  given  in  the  following  table : 

SOLAR  ECLIPSE,  July  18,  I860.— SOUTHERN  LIMIT. 


Greenwich 
Mean  Time. 

Latitude. 
* 

Long.  W.  from 
Greenwich. 

H 

0\520 

-f  16°    5' 

102°  40' 

0  .6 

21    32 

88    31 

0.8 

25      6 

76    37 

1.0 

26    36 

69      2 

1  .2 

27    17 

63      9 

1  .4 

27    27 

58    14 

1  .6 

27    15 

53    57 

1.8 

26   47 

50      9 

2  .0 

26     4 

46   43 

2  .2 

25      9 

43    33 

2  .4 

24     3 

40    34 

2  .6 

22   48 

37    45 

2.8 

21      5 

34    33 

3  .0 

19      9 

31    25 

3  .2 

16   41 

27    50 

3  .4 

14    14 

24    39 

3  .6 

11      9 

20    44 

3.8 

8      5 

16    55 

4  .0 

+    4     3 

11    46 

4.2 

—    0    39 

5    17 

4  .346 

-14    47 

339    30 

314.  We  have  applied  the  preceding  method  only  to  the  deter- 
mination of  the  extreme  limits  of  the  penumbra,  which  may  be 
designated  as  the  extreme  limits  of  partial  eclipse.  The  same 
method  will  determine  the  northern  and  southern  limits  of  total 
or  annular  eclipse,  by  employing  the  value  of  I  for  the  total 
shadow — that  is,  for  interior  contacts.  The  latter  are,  indeed, 
more  important,  practically,  than  the  former,  and  therefore  in 


CURVE    OF   CENTRAL    ECLIPSE.  491 

special  cases  somewhat  greater  precision  might  be  desired  than 
has  been  observed  in  the  preceding  example.  In  any  such  caser 
recourse  may  be  had  to  the  rigorous  method  of  Art.  311.  Since 
the  limits  of  total  or  annular  eclipse  often  include  but  a  very 
narrow  belt  of  the  earth's  surface,  extending  nearly  equal 
distances  north  and  south  of  the  curve  of  central  eclipse, 
they  may  be  derived,  with  sufficient  accuracy  for  most  purposes, 
from  this  curve,  by  a  method  which  will  be  given  in  Art.  320. 

The  curve  upon  which  any  given  degree  of  obscuration  can 
be  observed  may  also  be  computed  by  the  preceding  method.  It 
is  only  necessary  to  substitute  A  for  ?,  and  to  give  J  a  value  cor- 
responding to  D  according  to  the  equation  (529).  All  the  curves 
thus  found  begin  and  end  upon  the  curve  of  maximum  in  the 
horizon. 

Curve  of  Central  Eclipse. 

315.  To  find  the  curve  of  central  eclipse  upon  the  surface  of  the 
earth. — This  curve  contains  all  those  points  of  the  surface  of  the 
earth  through  which  the  axis  of  the  cone  of  shadow  passes.  The 
problem  becomes  the  same  as  that  of  Art.  298  upon  the  suppo- 
sition that  the  shadow  is  reduced  to  a  point — that  is,  when 
I  —  if  =  0,  and,  consequently,  by  (493), 

$  =  x  r)=y 

Hence,  putting 


the  equations  (502)  to  (508)  are  reduced  to  the  following  ex- 
tremely simple  ones,  which  are  rigorously  exact: 

sin  /9  sin  f  =  x 
sin  /?  cos  Y  —  yl 
c  sin  C  =  yl 
c  cos  C  —  cos  /9 
cos  ft  sin  $  =  x  (539) 

cos  ^j  cos  #  =  c  cos  ((7  -f  c?,) 
sin  ft=  c  sin  ((7  -f-  d^ 

tan  tf, 
tan  9?  —  -  &  =  f^  —  7? 

1/(1  —  ee) 

It  will  be  convenient  to  prepare  the  values  of  yl  for  the  prin- 
cipal hours  of  the  eclipse ;  and  then  for  any  given  time  Staking 
the  values  of  x,  yv  dl9  /^,  from  the  eclipse  tables,  these  equations 
determine  a  point  of  the  curve. 


492  SOLAR    ECLIPSES. 

316.  The  extreme  times  between  which  the  solution  is  possible, 
or  the  beginning  and  end  of  central  eclipse  upon  the  earth,  are 
found  as  follows.  At  these  instants  the  axis  of  the  shadow 
is  tangent  to  the  earth's  surface,  and  the  central  eclipse  is 
observed  at  sunrise  and  sunset  respectively.  Hence,  Z  being  the 
zenith  distance  of  the  point  Z,  we  have  cos  Z  =  0,  or,  by  (503), 
d=  0,  whence,  by  (499), 


or 


which  is  equivalent  to  putting  sin  /9  =  1,  or  cos  ft  =  0,  in  the 
first  two  equations  of  (539),  so  that  we  have 

sin  Y  =  #,  cos  Y  =  yl 

Let  xf  and  y/  denote  the  mean  hourly  changes  of  x  and  y^  com- 
puted by  the  method  of  Art.  296.  Let  the  required  time  of 
beginning  or  ending  be  denoted  by  T=  T0-\-  r,  T0  being  an 
arbitrarily  assumed  epoch  ;  then,  if  (x)  and  (y~)  are  the  values  of 
x  and  yl  taken  for  the  time  Tw  we  have  for  the  time  jT, 

sin  Y  =  O)   +  ^ 
cos  Y  =  (y,)  -f  y/r 

Let  w,  Mj  n,  N,  be  determined  by  the  equations 

m  sin  M  =  (x)  n  sin  JV  =  xr 

m  cos  M  =  (yt)  n  cos  JV  =  y{ 

then,  from  the  equations 

sin  Y  =  m  sin  M  -\-  n  sin  N.  r 
cos  y  =  m  cos  Jf  -f  n  cos  JV.  T 

we  deduce,  in  the  usual  manner, 

sin  (Y  —  -^7)  —  m  8i 

cos  (y  —  JV)  =  m  cos  (JkT  —  JV^)  -j- 


or,  putting  ^  —  r  —  -Z\T,  the  solution  is 

V) 

va  /•  }W  __    A7"^ 

(541) 


sin  4  =  m  sin  ( J/  —  JV) 

cos  4*        m  cos  (M  —  JV) 


CURVE    OF    CENTRAL    ECLIPSE.  493 

where  cos  ^  ls  to  be   taken  with   the   negative   sign   for  the 
beginning  and  with  the  positive  sign  for  the  end. 

To  find  the  latitude  and  longitude  of  the  extreme  points  cor- 
responding to  these  times,  we  have,  in  (539),  cos  ^5  =  0,  sin  /?  =  1, 
and,  therefore,  C=  90°,  c  =  cos  Y  •  hence,  taking  f  =  N+  $, 

cos  ^sin  #  —       sin  Y 

cos  <f>^  cos  #  =  —  cos  Y  sin  dl 

sin  <fi  =       cos  Y  cos  ^  (542) 

tan 
tan  <p  - 


317.  To  find  the  duration  of  total  or  annular  eclipse  at  any  point  of 
the  curve  of  central  eclipse. — This  is  readily  obtained  from  numbers 
which  occur  in  the  previous  computations.  Let  T=the  time 
of  central  eclipse,  t  =  the  duration  of  total  or  annular  eclipse, 
then  T'=  T  ^  %t  is  the  time  of  beginning  or  end.  Let  x  and 
y  be  the  moon's  co-ordinates  for  the  time  T;  £  and  rj  those  of 
the  point  on  the  earth  at  this  time  ;  a;',  y,  £',  i/,  the  hourly  in- 
crements of  these  quantities ;  then,  at  the  time  T'  we  have,  by 
(491), 

(I  -  1C)  sin  Q  =  x  +  iaft  -  (£  =p  K'O 
(/  —  f  C)  cos  Q  =  y  q;  i  y't  —  (7  T  }  VO 

But  we  here  have  x  =  £9  y  =  y9  and  we  may  put  £  =  f t  =  cos  jS, 
whence 

(Z  —  f  cos  /3)  sin  §  =  T  (#'  —  ^')  — 

21 


For  the  values  of  £'  and  tf  we  have,  with  sufficient  precision, 
since  t  is  very  small, 

f'=A.'(-y6in<*  +  co9/Jco8d) 
Tf)'=fj.'x  sin  6? 

Hence,  by  (511)  and  (512),  we  find,  very  nearly, 

x'  —  £'  =  cf  —  ,c/  cos  d  cos  /?  =  d  —  /  cos  ft 

' 
If,  therefore,  we  put 

L  =  l  —  i  cos  &  a  =  c'  —  /  cos  ft  (543) 


494  SOLAR    ECLIPSES. 

we  have 

L  sin  Q  =  —  L  cos  Q  =  — 

where  we  omit  the  double  sign,  since  it  is  only  the  numerical 
value  of  t  that  is  required.  Hence,  we  have,  for  finding  t,  the 
equations 


the  last  equation  "being  multiplied  by  3600,  so  that'it  now  gives 
t  in  seconds. 

The  value  of  cos  ft  is  to  be  taken  from  the  computation  of  the 
central*  curve  for  the  given  time  JJ  and  I,  log  i,  log/,  c',  6',  from 
our  eclipse  tables. 

318.  To  find  where  the  central  eclipse  occurs  at  noon.  —  In  this  case 
we  have,  evidently,  x  —  0,  and  hence,  in  (539), 

sin  0  =  y±  (545) 

by  which  ft  is  to  be  found  from  the  value  of  yl  which  corresponds 
to  the  time  when  x  =  0.  We  then  have  C  =  ft,  c  =  1,  #  =  0, 
and  therefore  the  required  point  is  found  by  the  formulae 

Vl=ft  +  dl  a>=^  (546) 

in  which  d^  and  ^  are  taken  for  the  time  when  x  =  0. 

319.  The  formula  (539),  (545),  and  (546)  are  not  only  extremely 
simple,  but  also  entirely  rigorous,  and  have  this  advantage  over 
the  methods  commonly  given,  that  they  require  no  repetition  to 
take   into   account  the   true   figure  of  the   earth.     It   may  be 
observed  here  that  the  accurate  computation  of  the  central  curve 
is  of  far  greater  practical  importance  than  that  of  the  limiting 
curves  before  treated  of. 

The  formulae  (541)  must  be  computed  twice  if  we  wish  to 
obtain  the  times  of  beginning  and  end  with  the  greatest  pos- 
sible precision  ;  for,  these  times  being  unknown,  we  shall  have 
at  first  to  employ  the  values  of  x'  and  y'  for  the  middle  of  the 
eclipse,  and  then  to  take  their  values  for  the  times  obtained  by 
the  first  computation  of  the  formulae.  With  these  new  values  a 
second  computation  will  give  the  exact  times. 


CURVE    OF    CENTRAL    ECLIPSE. 


495 


EXAMPLE. — To  compute  the  curve  of  central  and  total  eclipse 
in  the  eclipse  of  July  18,  1860. 

It  is  convenient  first  to  prepare  the  values  of  y^  =  —  for  the 

i°i 

principal  hours  of  the  eclipse,  as  well  as  its  mean  hourly  differ- 
ences. With  the  value  log  pl=  9.99873  we  form,  from  the  values 
of  y  given  Vn  +h^  table  p.  454,  the  following  table : 


Gr.  T. 

ri 

y\ 

0* 

+  0.91972 

—  0.16095 

1 

.75896 

114 

2 

.59782 

132 

3 

.43633 

149 

4 

.27450 

166 

£ 

.11237 

182 

To  find  the  times  of  beginning  and  end  we  may  assume  TQ=  2*; 
and  for  this  time  we  have 


(x  )  =  m  sin  M  =  —  0.08124 
(yt)  =  m  cos  M=  -\-  0.59782 
whence  log  m  =  9.78054 

M=       352°  15' 40" 


x'=  n  sin  N=  -f  0.5453 
y/  =  n  cos  N  =  —  0.1613 
log  n  —       9.7'548 
N=       106°28'.7 


Employing  but  four  decimal  places  in  the  logarithms  for  a  first 
approximation,  we  find,  by  (541), 


mcos(Jf 


=  +  0*.435 
=  q=  1  .468 


^  =  _  l  .033 
ra  —  +  1  .903 

Approximate  time  of  beginning  —  2*  —  1*.033  =  0*.967 
"  il          end  =2+1  .903  =  3  .903 

Taking  now  x'  and  yj  for  these  times  respectively,  and  re- 
peating the  computation,  we  have 


496 


SOLAR   ECLIPSES. 


Beginning. 

End. 

of  =  n  sin  N 

-f  0.54531 

4-  0.54525 

yf  —  n  cos  N 

—  0.16113 

-  0.16164 

log  ft 

9.75482 

9.75489 

N 

106°  27'  42" 

106°  30'  45" 

m  cos  (M  —  N) 

4-  0».4349 

4-  0*.4357 

n 

cos  4 

1  4684 

4-  1  .4685 

J.    •TTvlCT^r 

n 

T 

0  .9665 

3  .9042 

4 

213°  23'  12" 

326°  37'  40" 

For  the  latitude  and  longitude  of  the  points  of  beginning  and 
end,  we  now  take  7*  =  N  +  ^  and  with  the  values  of  dl  and 
A*!  (PP-  455,  464)  for  the  above  computed  times,  we  have 


Beginning. 

End.  ' 

319°  50'  54" 
21   1  15 
13   1  1 

73°  8'  25" 
21  0  0 
57  5  3 

whence,  by  (542), 


Local  App.  Time  =  & 


45°  36'  50" 
126      3     8 


15°  45'  34" 
320    53    9 

6*  24*8 


For  the  series  of  points  on  the  curve  we  take  the  times  1*.0, 
1A.2,    1*.4  .......  3^.6,    3\8,   which    are    embraced   within   the 

extreme  times  above  found,  and  proceed  by  (539).     Thus,  for 

2A.0  we  have 

T  2*. 

x  =  sin  ft  sin  r        -  0.08124 
yl  =  sin  /3  cos  r 
log  sin  p 

log  cos  p  =  log  c  cos  C 

log  yl  =  log  c  sin  C 

log  c 

C 


+  0.59782 
9.78054 
9.90173 
9.77657 
9.99856 
36°  51'  21" 
21  0  49 


CURVE    OF    CENTRAL    ECLIPSE. 


497 


log  x  =  log  cos  ^  sin  # 
log  c  cos  ((7  -f-  JJ  =  log  cos  ^j  cos  ft 
log  c  sin  ((7  -j-  rfj)  =  log  sin  ^ 


App.  Time  =  &  in  time, 


57°  52'  10" 
wS.90977 
9.72435 
9.92636 
351°  17'  13" 
28    31  12 
37    13  59 
57    39  20 
23*  25"  8'.8 


For  the  duration  of  totality  at  this  point,  we  take  from  pp.  454, 
464,  465, 

I  =  _  0.009082  V  =  +  0.1532 

log  i  =       7.6608  c'  =  +  0.6011 

log/=       9.3883 

and  hence,  with  log  cos  ft  —  9.9017  above  found,  we  obtain,  by 
(543), 

L  =  —  0.012734  a  =  +  0.4061 

and,  by  (544),  disregarding  the  negative  sign  of  Z/, 
*  =  211-.3  =  3-  31'.3 

For  the  place  where  the  central  eclipse  occurs  at  noon,  we  nnd 
that  x  =  0  at  the  time  T=  2M49,  at  which  time  we  have 


=  sin  0    -f  0.57378 

35°  0'  53" 
21   0  45 
56   1  38 
56   6  57 
30  45  18 


The  whole  curve  may  be  traced  through  the  points  given  in  the 
following  table : 


VOL.  I.— 32 


498 


SOLAR    ECLIPSES. 


SOLAR  ECLIPSE,  July  18,  I860.— CURVE  OF  CENTRAL  AND  TOTAL  ECLIPSE. 


Greenwich 

Latitude. 

Long.  W.  from 
Greenwich. 

App.  Local 
Time. 

Duration  of 

Mean  Time. 

A 

Totality. 

r 

u 

# 

0*.967 

45°  36'.4 

126°    3M 

16*  27TO.9 

1  .0 

50    37.8 

113    11.6 

17   21  .3 

2m    1'.5 

1.2 

57    16.2 

89    14  .6  |    19     9  .1 

2    35.1 

1  .4 

59    29.1 

72    52.8 

20   26  .6 

2    55.8 

1  .6 

59    55.1 

59      5.2 

21    33  .7 

3    11.4 

1  .8 

59    11.6 

47    16.6 

22   33  .0 

3    23.1 

2  .0 

57    39.3 

37    14.0 

23   25  .1 

3    31.3 

2  .149 

56      7.0 

30    45.3 

0     0  .0 

3    34.7 

2  .2 

55    31.5 

28    42.6 

0   11  .2 

3    36.2 

2  .4 

52    56.9 

21    25.1 

0   52  .4 

3    38.0 

2.6 

50      0.9 

15      3.9 

1   29  .8 

3    36.4 

2.8 

46    46  3 

9    21.8 

2     4  .6 

3    32.0 

3  .0 

43    13.6 

4      2.2 

2   37  .9 

3    24.6 

3.2 

39    20.7 

358    47.1 

3   10  .9 

3    14.4 

3.4 

35      1.6 

353    12.5 

3   45  .3 

3      1.1 

3  .6 

30      1  .5 

346    35.4 

4   23  .7 

2    43.5 

3i  B 

23    28.5 

336    44.1 

5   15  .1 

2    18.5 

3  .904 

15    45.6 

320    53.2 

6   24  .8 

Northern  and  Southern  Limits  of  Total  or  Annular  Eclipse. 

320.  To  find  the  northern  and  southern  limits  of  total  or  annular 
eclipse. — As  already  remarked  in  Art.  314,  these  limits  may  be 
rigorously  determined  by  the  method  of  Art.  311,  by  taking 
I  =  the  radius  of  the  umbra  (i.e.  for  interior  contacts) ;  but  I  here 
propose  to  deduce  them  from  the  previously  computed  curve  of 
central  eclipse.  This  radius  I  is  assumed  to  be  so  small  that  we 
may  neglect  its  square,  which  can  seldom  exceed  .0003,  and  this 
degree  of  approximation  will  in  the  greater  number  of  cases 
suffice  to  determine  points  on  the  limits  within  2'  or  3',  which  is 
practically  quite  accurate  enough. 

The  two  limiting  curves  of  total  or  annular  eclipse,  then,  lie 
so  near  to  the  central  curve  that  the  value  £t  =  cos  /9,  for  a  given 
time  y,  already  found  in  the  computation  of  the  latter  curve, 
may  be  used  for  the  former  in  the  approximate  equation  which 
determines  Q.  We  can,  therefore,  immediately  find  Q  by  (535), — i.e. 


LIMITS    OF    TOTAL    OR   ANNULAR    ECLIPSE.  499 


tan  /  =  —  cos 


(547) 
tan  (Q  —  IE)  =  tan  (45°  +  «/)  tan  \E 


) 

1 

) 


where  /,  e,  and  E  are  to  be  taken  from  the  eclipse  tables  for  the 
time  T. 

The  co-ordinates  of  the  point  on  the  central  curve  correspond- 
ing to  the  time  T  being  £  =  x  and  yt  =  r^  (Art.  315),  those  for 
a  point  on  the  limiting  curve  may  be  denoted  by  x  +  dx  and 
y\  +  tyr  These  being  substituted  for  £  and  ^  in  the  equations 
(499),  we  have 

dx  =  —  (I  —  XJ  sin  Q  dyl=  —  (l  —  tC,)  cos  Q 

where  in  the  expression  for  dyl  we  omit  the  divisor  pl9  as  not 
appreciably  changing  the  value  of  so  small  a  term. 

Let  <pu  $,  co  be  taken  from  the  computation  of  the  central 
curve  for  the  time  T7,  and  let  <pl  +  dyl9  co  +  ^,  be  the  cor- 
responding values  of  <pl  and  co  for  the  point  on  the  limit  for 
the  same  time.  Then,  by  differentiating  (500),  observing  that 
d&  =  —  dco,  we  have 


cos  pj  cos  #  du  -f-  sin  ^t  sin  #  d<p 2  =  —  <£r 
cos  ^  sin  #  dw  —  sin  ^  cos  #  rf^  =  —  dyt  sin  rfj  -}-  ^Ct  cos  f/x 
cos  <p^d<pi  =       dyl  cos  ^  -f  dCj  sin  6?t 

whence,  by  eliminating  d^l  and  substituting  f  t  for  its  value  given 
by  the  third  equation  of  (497),  we  find 

C4  cos  (f>i  da*  =  —  dx  (cos  <pt  cos  rfj  -j-  sin  <pl  cos  ^  sin  d^ 

—  dyt  sin  ^t  sin  »> 
Cj^rrr  —  ^jc    sin  ^  sin  rfj  -f-  dyl  cos  t> 

Hence,  substituting  cos  ft  for  f n 
?  —  i  cos  /? 


cos/9 


(cos  t9  sin  Q  sin  rf,  4-  sin  #  cos  $)  tan 


.    ?  —  i  cos  p   .     _ 

—  sin  §  cos  d. 
cos/3 

dy  =  —  —  —  (sin  &  sin  Q  sin  d±  —  cos  &  cos  0) 

cos  ? 


500  SOLAR    ECLIPSES. 

These  values  are  yet  to  be  divided  by  sin  V  to  reduce  them  to 
minutes  of  arc.     It  will  be  convenient  to  put 


p. 


sin  1'  sin  1' 


_  I  _  |  cos  ft  _     Z' 


cos  /9  sin  1'       cos 


(548) 


in  which  £',  £',  and  A  will  be  expressed  in  minutes. 

We  may  in  practice  substitute  d<p  for  dtp^  within  the  limits  of 
accuracy  we  have  adopted  ;  for  we  find,  from  the  equations  on 
p.  457, 

dtp,          cos2  <p  1  —  ee  sin2  <p 


where  the  multiplier  of  d(pl  cannot  differ  more  from  unity  than 
]/(!  —  ee)  does,  —  i.e.  not  more  than  0.00335:  so  that  the  substitu- 
tion of  one  for  the  other  will  never  produce  an  error  of  1'  so  long 
as  d<p±  is  less  than  5°. 

Finally,  adapting  the  values  of  da)  and  dtp  for   logarithmic 
computation,  by  putting 

h  sin  H  =  cos  Q 
h  cos  H  =  sin  Q  sin  dt 
we  have 

dot  =  A  [h  cos  (#  —  H)  tan  ^  -f-  sin  §  cos  dj 


(549) 

3  (#  —  H)  tan  yv  -f-  sin  §  cos  ^J 
sin 


The  formula  (547)  give  two  values  of  Q  differing  180°.  The 
second  value  will  evidently  give  the  same  numerical  values  of 
dto  and  efy,  but  with  opposite  signs  ;  and  therefore  we  may  com- 
pute the  equations  (549)  with  only  the  acute  value  of  ft  and  then 
the  longitude  and  latitude  of  a  point  on  one  of  the  limits  are 

to  -\-  dm,  <f>  -\-  d<f> 

and  those  of  a  point  on  the  other  limit  are 

at   —  du),  <p  —  d<p 

The  first  of  these  limits  will  be  the  northern  in  the  case  of 
total  eclipse,  but  the  southern  in  the  case  of  annular  eclipse, 
observing  always  to  take  I  with  the  negative  sign  for  total  eclipse, 
as  it  comes  out  by  the  formulae  (487)  and  (489). 


LIMITS    OF    TOTAL    OR   ANNULAR    ECLIPSE. 


501 


It  is  evident  that  this  approximate  method  is  not  accurate 
when  cos  ft  is  very  small,  that  is,  near  the  extreme  points  of  the 
curves;  and  it  fails  wholly  for  these  points  themselves,  since 
cos  /9  is  then  zero  and  the  value  of  A  becomes  infinite.  These 
extreme  points,  however,  are  determined  directly  in  a  very 
simple  manner  by  the  formulae  (536),  (537),  (538),  combined  with 
(519),  by  employing  in  (536)  and  (537)  the  value  of  I  for  interior 
contacts;  and  it  is  with  these  formulae,  therefore,  that  the  com- 
putation of  the  limits  of  total  or  annular  eclipse  should  be  com- 
menced. 

EXAMPLE. — Find  the  northern  and  southern  limits  of  total 
eclipse  in  the  eclipse  of  July  18,  1860. 

First.  To  find  the  extreme  points. — The  values  of  br  and  cf  for 
exterior  contacts,  from  which  the  values  of  E  on  p.  465  are 
derived,  differ  so  little  from  those  for  interior  contacts  that  in 
practice,  unless  extreme  precision  is  required,  we  may  dispense 
with  the  computation  of  the  latter.  For  our  present  example, 
therefore,  taking  the  value  of  .Z£for  T0=  2h  and  the  mean  value 
of  log  e,  as  in  the  computation  of  the  extreme  points  of  the 
southern  limit  for  the  penumbra,  p.  487,  together  with 

I  =  —  0.0091 
we  find,  by  (537),  for  the  northern  limit, 


log  m  =  9.7854 
log  n  =  9.7553 

and  for  the  southern  limit, 


Jf=352°33'.6 
N  =  106    27'.0 


log  m  =  9.7731 
log? 

Hence,  by  (538), 


M  =r 


log  n  =  9.7542 


Northern  Limit. 


55'.0 
27.0 


Southern  Limit. 


First  Point. 

Last  Point. 

First  Point. 

Last  Point. 

213°  54'.3 

0*.976 

326°    5'.7 
3*.892 

212°39'.0 
0».951 

327°  21'.0 
3*.917 

Taking  f  =  N-\-  ^,  and  the  values  of  d{  and  ^  for  these  times 
respectively,  with  log  ^  =3  9.9987,  we  find,  by  (518)  and  (519), 


502 


SOLAR    ECLIPSES. 


log  tan 


320°  21'.3 

72°  32'.7 

319°  6'.0 

73°  48  .0 

7i9.9170 

0.5012 

W9.9363 

0.5355 

21°  1'.2 

21°  O'.O 

21°  T.2 

21°  O'.O 

246  31.7 

96  26.7 

247  26.7 

95  57.7 

13   9.6 

56  54.1 

12  47.1 

57  16.6 

126  37.9 

320  27.4 

125  20.4 

321  18.9 

46   7.7 

16  21.6 

45   2.8 

15  11.4 

Second.  To  find  a  series  of  points  between  these  extremes,  by 
the  aid  of  the  curve  of  central  eclipse,  we  assume  the  same  series 
of  times  as  in  the  computation  of  that  curve,  and  proceed  by 
(547),  (548),  and  (549) ;  to  illustrate  the  use  of  which  I  add  the 
computation  for  T=  2h  in  full.  From  the  computation,  p.  496, 
we  have 

log  cos  p  9.9017 

log  tan  ^t          0.1970 

351°  17'.2 
21  0.8 
37  14.0 
57  39.3 


Then,  by  (547), 


(p.  465)  log£ 

9.5953 

log  cos  p 

9.9017 

log  tan  / 

9.4970 

v' 

17°  26'.0 

\E 

7      8.7 

log  tan  (45°  -f  v') 

0.2823 

log  tan  \  E 

9.0982 

logtan(§—  %E) 

9.3805 

Q—fS 

13°  30'.3 

Q 

20    39.0 

Hence,  by  (549), 

By  (548), 

I 

log  I 
log  I' 
log  i 
log  i' 
i' 
I'  sec  y? 

a 


—  0.009082 
7i7.9582 
nl.4945 

7.6608 
1.1971 
15'.74 

—  39.16 
-  54.90 


log  cos  Q  =  log  A  sin  H 

log  sin  Q  sin  dl=log  h  cos  H 

logh 

H 


9.9712 
9.1020 
9.9751 
82°  18'.2 
268    59.0 


LIMITS    OF   TOTAL    OR    ANNULAR    ECLIPSE.                        503 

log  A 

nl.7396 

log  A 

nl.7396 

log  h 

9.9751 

logfc 

9.9751 

log  cos  (#  —  H) 

w8.2490 

log  sin(#  —  H) 

n9.9999 

log  tan  <fv 

0.1970 

log  dtp 

1.7146 

log  (1) 

0.1607 

dy> 

+  51'.83 

log  A 

nl.7396 

log  sin§  cos  dl  . 

9.5175 

log  (2) 

nl.2571 

(1) 

+    1'.45 

(2) 

—  18  .08 

dto 

-  16  .63 

Hence,  for  the  time  T=  2*,  we  have  the  two  points, 


at  •+:  da) 
<p  H-  d(f> 

N.  Limit. 

S.  Limit. 

36°  57/4 
58    31.1 

37°  30'.6 
56    47.5 

SOLAR  ECLIPSE,  July  18,  1860. 
Northern  Limit  of  Total  Eclipse.  Southern  Limit  of  Total  Eclipse. 


Gr.  T. 

Latitude. 
4 

Longitude. 
cj 

0*.976 
1  .0 
1  .2 
1  .4 
1  .6 
'1  .8 

46°  8' 
50  18 
57  47 
60  13 
60  46 
60   4 

126°  38' 
116  27 
90  57 
74   0 
59  40 
47  23 

2  .0 
2.2 
2  .4 
2.6 

2.8 

58  31 
56  21 
53  43 
50  43 
47  24 

36  57 
28   9 
20  40 
14  12 

8  44 

3  .0 
3.2 
3  .4 
3  .6 

3.8 
3  .892 

43  47 
39  49 
35  25 
30  18 
23  31 
16  22 

3   1 

357  43 
352   6 
345  23 
335   8 
320  27 

Gr.  T. 

Latitude. 

Longitude. 
d 

0*.951 
1.0 
1  .2 
1  .4 
1  .6 
1  .8 

45°  3' 
50  57 
56  45 
58  45 
59   4 
58  19 

125°  20' 
109  56 
87  33 
71  46 
58  31 
47  11 

2  .0 
2  .2 
2  .4 
2  .6 
2\8 

56  48 
54  42 
52  11 
49  19 
46   9 

37  31 
29  16 
22  10 
15  56 
10  39 

3.0 
3  .2 
3  .4 
3  .6 

3.8 
3  .917 

42  41 
38  52 
34  38 
29  45 
23  26 
15  11 

5   3 

359  51 
354  20 
347  48 
338  20 
321  19 

504  SOLAR    ECLIPSES. 

321.  The  curves  above  computed  are  all  exhibited  in  the  fol- 
lowing chart. 


SOLAB  EOLIPSE-July  18,  1860, 


For  the  construction  of  such  charts,  on  even  a  much  larger 
scale,  the  degree  of  accuracy  with  which  our  computations 
have  been  made  is  far  greater  than  is  necessary,  and  many 
abridgments  may  be  made  which  will  readily  occur  to  the 
skilful  computer.* 


*  For  a  graphic  method  of  constructing  eclipse  charts,  see  a  paper  by  Mr. 
CHAUNCEY  WRIGHT,  Proceedings  of  the  Am.  Association  for  the  Adv.  of  Science,  8th 
meeting  (1854),  p.  55. 


PREDICTION   FOR   A   GIVEN   PLACE.  505 

Prediction  of  a  Solar  Eclipse  for  a  Given  Place. 

322.  To  compute  the  time  of  the  occurrence  of  a  given  phase  of  a 
solar  eclipse  for  a  given  place.  —  The  given  phase  is  expressed  by  a 
given  value  of  J,  and  we  are  to  find  the  time  when  this  value 
and  the  co-ordinates  of  the  given  place  satisfy  the  conditions 
(485).  This  can  only  be  done  by  successive  approximations. 

Let  it  be  proposed  to  find  the  time  of  beginning  or  ending  of 
the  eclipse  at  the  place.  The  phase  is  then  A  =  I  —  if,  and  we 
must  satisfy  the  equations  (491).  Let  TQ  be  an  assumed  time, 
and  T=  T0  -f  r  the  required  time.  Let  x,  y,  x',  y',  d,  ?,  log  i,  be 
taken  from  the  eclipse  tables  (p.  454)  for  the  time  T0.  Assuming 
that  x  and  y  vary  uniformly,  their  values  at  the  time  T  are 
x  -f  X'T  and  y  -f  y'r.  The  co-ordinates  of  the  place  at  the  time 
T0  are  found  by  (483)  or  (483*),  in  which  //  is  the  sidereal  time 
at  the  place.  Putting 


in  which  CD  is  the  west  longitude  of  the  place  and  //1  may  be  taken 
from  the  table  (p.  455)  for  the  time  T0,  the  formulae  become 

A  sin  B  =  p  sin  <f  £  =  p  cos  <f>'  sin  $  ^ 

A  cos  B  =  p  cos  yr  cos  #      7?  —  A  sin  (B  —  d~)  >  (550) 

Z=Acos(B  —  d)  ) 

Let  ?',  tf  denote  the  hourly  increments  of  £  and  rj  ;  then,  assuming 
that  these  increments  also  are  uniform,  the  values  of  the  co-ordi- 
nates at  the  time  T  are  £  +  £'r  and  ^  -j-  Y/T.  The  values  of  £' 
and  rf  are  found  by  the  formulae  (p.  462) 

$  =  ft!  p  cos  <p'  cos  # 

V=//£sin  d  —  d'Z 

in  which  //  and  d'  are  the  hourly  changes  of  //  and  d  multiplied 
by  sin  V.  The  rate  of  approximation  will  not  be  sensibly 
affected  by  omitting  the  small  term  d'^,  and  the  formulae  for  k' 
and  rf  may  then  be  written  as  follows  : 

$'  =  t*'  A  cos  B  i)'  =  p!*Bmd  (551) 

Put 

L  =  i  —  i: 

then,  neglecting  the  variation  of  this  quantity  in  the  first  ap- 
proximation, the  conditions  (491)  become,  for  the  time  JJ 

L  sin  Q  =  x  —  $  -f  (xf  —  £')  T 
L  cos  Q  =  y  —  ^  -j-  (yf  —  V)  r 


506  SOLAR   ECLIPSES. 

Let  the  auxiliaries  m,  If,  n,  and  N!OQ  determined  by  the  equa- 
tions 

m  sin  M  —  x  —  £  n  sin  N  =  xf  —  £' 

m  cos  M  =  y  —  y  n  cos  N  =y'  —  if 

then,  from  the  equations 

L  sin  Q  =  m  sin  Jlf  -j-  n  sin  JV.  r 
_L  cos  Q  =  m  cos  3/  -f-  n  cos  JV.  r 
we  deduce 

L  sin  (Q  —  N~)  =  m  sin  (M  —  N) 

L  cos  (  Q  —  N)  =  m  cos  (M  —  JV)  -f-  nr 

Hence,  putting  $=  Q  —N,we  have 
m  sin  ( M  —  N) 


sm  ^  = 


_  j^  cos  4-       m  cos  ( M  —  JV) 
w  w 


(553) 


by  which  r  is  found.  Since  the  first  of  these  equations  does  not 
determine  the  sign  of  cos  ^  the  latter  may  be  taken  with  either 
the  positive  or  the  negative  sign.  We  thus  obtain  twro  values 

of  T=  T0  +  r,  the  first  given  by  the  negative  sign  of  - 

n 

being  the  time  of  beginning,  and  the  second  given  by  the  posi- 
tive sign  being  the  time  of  ending  of  the  eclipse  at  the  place. 

For  a  second  approximation,  let  each  of  the  computed  times 
(or  two  times  nearly  equal  to  them)  be  taken  as  the  assumed 
time  ro,  and  compute  the  equations  (550),  (551),  (552),  (553)  for 
beginning  and  end  separately. 

The  first  approximation  may  be  in  error  several  minutes,  but 
the  second  will  always  be  correct  within  .  a  few  seconds,  and, 
therefore,  quite  as  accurate  as  can  be  required  ;  for  a  perfect 
prediction  cannot  be  attained  in  the  present  state  of  the  Ephe- 
merides. 

The  formula  for  r  may  also  be  expressed  as  follows  : 

m   sin  (M  —  ^V  —  4) 


n  sn 


which  in  the  second  approximation  will  be  more  convenient 
than  the  former  expression  ;  but  when  sin  ^  is  very  small  it  will 
not  be  so  precise. 


PREDICTION  FOR  A  GIVEN  PLACE.  507 

If  we  put 

t  =  the  local  mean  time  of  beginning  or  end, 
we  have 

t=T0+T-a>. 

323.  The  prediction  for  a  given  place  being  made  for  the 
purpose  of  preparing  to  observe  the  eclipse,  it  is  necessary  also 
to  know  the  point  of  the  sun's  limb  at  which  the  first  contact  is 
to  take  place,  in  order  to  direct  the  attention  to  that  point.  This 
is  given  at  once  by  the  value  of 


which  is  the  angular  distance  of  the  point  of  contact  reckoned 
from  the  north  point  of  the  sun's  limb  towards  the  east  (Art.  295). 

The  simplest  nrethod  of  distinguishing  the  point  of  contact  on 
the  sun's  limb  is  (as  BESSEL  suggested)  by  a  thread  in  the  eye-piece 
of  the  telescope,  arranged  so  that  it  can  be  revolved  and  made 
tangent  to  the  sun's  limb  at  the  point.  The  observer  then,  by  a 
slow  motion  of  the  instrument,  keeps  the  limb  very  nearly  in 
contact  with  the  thread  until  the  eclipse  begins.  The  position 
of  the  thread  is  indicated  by  a  small  graduated  circle  on  the  rim 
of  the  eye-piece,  as  in  the  common  position  micrometer. 

This  method  is  applicable  whatever  may  be  the  kind  of 
mounting  of  the  telescope.  Nevertheless,  if  the  instrument  is 
arranged  with  motion  in  altitude  and  azimuth,  it  will  be  conve- 
nient to  know  the  angle  of  the  point  of  contact  from  the  vertex 
of  the  sun's  limb,  which  is  that  point  of  the  limb  which  is  nearest 
to  the  zenith.  The  distance  of  the  vertex  from  the  north  point 
of  the  limb  is  equal  to  the  parallactic  angle  which  being  here 
denoted  by  f,  is  found,  according  to  Art.  15,  by  the  formulae 

p  sin  f  =  cos  <p  sin  # 

p  cos  f  =  sin  <f>  cos  d  —  cos  y  sin  d  cos  # 

(where  we  have  put  p  for  sin  £  and  #  for  the  sun's  hour  angle). 
As  f  is  not  required  with  very  great  accuracy,  we  may  here  take 
[see  (494)] 

p  sin  Y  =  5  P  cos  f  =  y 

in  which  £  and  y  are  the  values  of  the  co-ordinates  of  the  place 
at  the  instant  of  contact.  But,  if  £  and  57  denote  the  values  at  the 
tune  JJ,,  we  must  take 

p  sin  Y  —  £  -f  £'*•  p  cos  f  =  y  -f  >/r  (554) 


508  SOLAR    ECLIPSES. 

in  which  we  employ  the  values  of  c,  37,  £',  r/,  and  r  furnished  by 
the  last  approximation.  We  then  have 

Angular  distance  of  the  point  of  contact  from  1  =  Q  —  f 

the  vertex  towards  the  east,  J  =  N  -j-  4,  —  ^ 

324.  To  find  the  instant  of  maximum  obscuration  for  a  given  place, 
mid  the  degree  of  obscuration.  —  At  the  instant  of  greatest  obscura- 
tion the  distance  J  of  the  axis  of  the  shadow  from  the  place  of 
observation  is  a  minimum.*  If  we  denote  the  required  time  by 
7\  =  T0  +  TJ,  the  equations  of  Art.  822  determine  rl  for  a  given 
value  of  J  if  we  substitute  J  for  L.  Denoting  the  value  of 
Q  —  N  for  this  case  by  $„  we  have,  therefore, 

A  sin  4/j  =  m  sin  (M  —  N) 

J  cos  4±  =  m  cos  (  M  —  N}  -f  nrt 

the  sum  of  the  squares  of  which  gives 

J2=  M2sin2(Jf  —  JV)  -f  [m  cos  (M  —  JST)  -f  wr,]2 

Since  m  and  J/  are  computed  for  the  time  Tw  and  N  is  sensibly 
constant,  the  term  m2sin2(M  —  N)  is  constant,  and  therefore  J 
is  a  minimum  when  the  last  term  is  zero,  that  is,  when 

m  cos  (M  —  N 


which  quantity  is  already  known  from  the  computation  of  (553). 
We  have,  also, 

A  =  ±:  m  sin  (M  —  N}  =  ±  L  sin  $  (557) 

in  which  the  sign  is  to  be  so  taken  as  to  make  J  positive.     The 
degree  of  obscuration  is  then  given  by  the  formula  (Art.  310), 


in  which  D  is  expressed  in  fractional  parts  of  the  sun's  diameter, 
and  L  and  L^  are  the  radii  of  the  penumbra  and  umbra  (the 

*  More  strictly,  L  —  J  is  a  maximum,  as  in  Art.  309 ;  but  we  here  neglect  the 
small  variation  of  L.  The  rigorous  solution  of  the  problem  may  be  obtained  from 
the  condition  (526)  P'  =  0 ;  but  the  above  approximation  is  sufficient  in  practice. 


PREDICTION    FOR    A    GIVEN    PLACE. 


509 


latter  being  negative)  for  the  place  of  observation.     From  (488) 
we  find,  by  putting  sec/=  1, 


and  hence 


in  which  k  =  0.2723. 

If  we  neglect  the  augmentation  of  the  moon's  diameter,  or, 
which  is  equivalent,  the  small  difference  between  L  and  £,  and 
put 


we  have 


D  = 


e  sin 


(559) 


where  the  lower  sign  is  to  be  used  when  sin  $  is  negative,  so 
that  D  is  always  the  numerical  difference  of  e  and  e  sin  i//.  In  this 
form  e  may  be  computed  for  the  eclipse  generally,  and  <$,  will  be 
derived  from  the  computation  for  the  penumbra  for  the  given 
place.  A  preference  should  be  given  to  the  value  of  ^  found 
from  the  computation  for  the  time  nearest  to  that  of  greatest 
obscuration,  which  is  usually  that  used  in  the  first  approximation 
of  Art.  322. 

EXAMPLE. — Find  the  time  of  beginning  and  end,  &c.,  of  the 
eclipse  of  July  18,  1860,  at  Cambridge,  Mass. 
The  latitude  and  longitude  are 


<p  =  42°  22'  49" 


=  71°  7'  25" 


For  this  latitude  we  find,  by  the  aid  of  Table  III.,  or  by  the 
formulae  (87), 


log  p  sin  <pf  =  9.82G44 


log  p  cos  <pf  =  9.86912 


With  the  aid  of  the  chart,  p.  504,  we  estimate  the  time  of  the 
middle  of  the  eclipse  at  Cambridge  to  be  not  far  from  1*.  Hence, 
taking  T0  =  1*  for  our  first  approximation,  we  take  for  this  time, 
from  the  eclipse  tables,  p.  454, 


x  =  _  0.6266 
y—  -f  0.7567 
d=  20°57'.4 


x'=  -f  0.5453 
y'=  —  0.1605 
i=  13°  31'.2 


I  =  0.5368 
log  i  —  7.66287 
log //'=  9.41799 


510 


SOLAR    ECLIPSES. 


Hence,  by  (550)  and  (551), 

//!  —  u  =  &—       302°23'.8 
B=         59   24.6 

%  =  _  0.6246 
rj  =  _j_  0.4844 

f '=:  -j-  0.1038 

if  =  —  0.0585 

and,  by  (552)  and  (553), 

m  sin  M  =  z  —  f  =  —  0.0020 

wi  cos  3/  =  y  —  T]  =  -f-  0.2723 

log  TO  =        9.4350 

J/"=       359°  34'.  7 
M—N=       256   34.1 
log  sin  4,  =      »9.6955 
log  cos  4,=       9.9387 


log£=       9.7853 

t£  =       0.0028 

=  Z  — t=       0.5340 


n  sin  JV=  z'—  f  '=  -f  0.4415 
n  cos  JV  =  y'  —  if  =  --•  0.  1020 
log  n  ==       9.6562 
JV=      103°0'.6 


^^    1.023 

—   0.883 
.or  +    1.163 
Approximate  time  of  beginning  =  OM17 
end  =2.163 


Taking  then  for  a  second  approximation  TQ=  OM2  for  begin- 
ning, and  7;=  2M6  for  end,  we  shall  find* 


5? 


M! 
I 

0 

£ 

^ 
logC 

i' 


Beginning. 

End. 

OM2 

2M6 

—  1.10642 

-f  0.00601 

+  0.89783 

+  0.57034 

+  0.54528 

-f-  0.54530 

—  0.16015 

-  0.16090 

20°  57'  45" 

20°  56'  53" 

0  19  8 

30  55  13 

0.53686 

0.53673 

289°  11'  43" 

319°  47'  48" 

—  0.69868 

-  0.47755 

-f  0.53915 

-f  0.42423 

9.66935 

9.88504 

+  0.06368 

+  0.14793 

—  0.06544 

—  0.04470 

*  The  values  of  x'  and  y'  here  employed  are  not  those  given  in  the  table  p.  455, 
but  their  actual  values  for  the  time  TQ,  as  given  in  the  table  of  x'  and  y'  on  p. 
464. 


PREDICTION  FOR  A  GIVEN  PLACE. 


511 


1C 

Beginning. 

End. 

0.00215 

0.00353 

L 

m  sin  M 

0.53471 

0.53320 

—  0.40774 

+  0.48356 

m  cos  M 

-f  0.35868 

+  0.14611 

log  m 

9.73484 

9.70342 

M 

311°  20'  16" 

73°  11'  15" 

n  sin  7V 

+  0.48160 

-f  0.39737 

71  COS  N 

-  0.09471 

—  0.11620 

logn 

9.69093 

9.61702 

^f  —  N 

101°    7'  32" 

106°  18'    0" 

210    12  44 

326    53  15 

4 

210    44     0 

328    49  56 

Jf  —  7V  —  4, 

31'  16" 

—      1°  56'  41" 

T 

-f  0*.0197 

-f-  0».0800 

f 

OM397 

2  .2400 

1 

0*   8TO  23* 

2*  14™  24- 

W 

4  44    30 

4  44    30 

Local  time,          t  \ 

AngleofPt.  of  Contact  from  ^ 
North  Pt.  of  the  sun  =  V 

19  23    53 
July  17. 

311°  51'  32" 

21   29    54 
July  17. 

75°    7'  56" 

A  third  approximation,  commencing  with   the   last   computed 
times,  changes  them  by  only  a  fraction  of  a  second. 

To  find  the  angular  distance  of  the  point  of  contact  from  the 
vertex  of  the  sun's  limb,  we  have  from  the  second  approximation, 
by  (554)  and  (555), 


Beginning. 

End. 

•      £' 

—  0.6974 

-  0.4658 

-\-  r/T  =  p  COSf 

+  0.5379 

+  0.4206 

r 

307°  38'.8 

312°  4'.5 

vertex  =  Q  —  y 

4    12.7 

123    3.4 

The  time  of  greatest  obscuration  is  best  found  from  the  first 
approximation,  which  gives,  by  (556), 


512  SOLAR    ECLIPSES. 


T,  =         1M40 

1*  8-  24'. 
^  =         4  44    30 
Local  time  of  max.  obscur.  =  t  =       20  23    54 

For  the  amount  of  greatest  obscuration  we  have,  also,  from 
the  first  approximation,  by  (557)  and  (558), 

L  =  0.5340  log  L  =    9.7275 

k  =  0.2723  log  sin  4  =  K9.6955 


k    =  0.2617  log  J  =    9.4230 

A-)  =  0.5234  J=    0.2649 

X  —  J  =    0.2691 


0.5234 

Or,  by  (559),  taking  as  constant  the  value  of  e  found  by  employ- 
ing the  mean  value  I  =  0.5367,  i.e. 

e  =  1.015 
\ve  have 

e  sin  4,  =  —  0.503 
D  =       0.512 

which  is  quite  accurate  enough. 

325.  Prediction  for  a  given  place  by  the  method  of  the  American 
Ephemeris.  —  This  method  is  based  upon  a  transformation  of 
BESSEL'S  formula  suggested  by  T.  HENRY  SAFFORD,  Jr.,  and,  with 
the  aid  of  the  extended  tables  in  the  Ephemeris,  is  somewhat 
more  convenient  than  the  preceding.  The  fundamental  equa- 
tion (490)  gives,  by  transposition, 

(x  -  £)'  =  (l  —  :  tan/)'  -  (y  -  ?)• 
the  second  member  of  which  may  be  resolved  into  the  factors 


or,  by  (494), 

b  =  I  -f  y  —  p  sin  <p'  (cos  d  -\-  Bin  d  tan  /) 

-f-  p  cos  <f>'  (sin  d  —  cos  d  ta-n  /)  cos 

c  =  I  —  y  -\-  p  sin  <f'  (cos  d  —  sin  d  tan/) 

—  p  cos  </>'  (sin  d  -f-  cos  d  tan  /)  cos 


PREDICTION   FOB   A  GIVEN   PLACE.  -513 

If  we  put 

A=x  B  =  l  +  y  C=  —  l  +  y 

E  =  cos  d  -f-  sin  d  tan/  =  cos  (d  —  /)  sec/ 
F  =  cos  d  —  sin  d  tan  /  =  cos  (d  +/)  sec/ 
G  =  sin  d  —  cos  d  tan/  =  sin  (d  —  /)  sec/ 
1T=  sin  d  +  cos  d  tan/  =  sin  (d  -f/)  sec/ 

all  of  which  are  independent  of  the  place  of  observation  and  are 
given  in  the  Ephemeris  for  each  solar  eclipse,  for  successive 
times  at  the  Washington  meridian,  we  shall  then  have  to  com- 
pute for  the  place 

a  =  x  —  $  =  A  —  p  cos  c'  sin  *  ^ 

b=  B—Ep  sin  ?'+  Gp  cos  /cos*  V    (560) 

) 


=  —  O  +  Fp  sin  ?'—  Hp  cos  /  cos  # 
and  the  fundamental  equation  becomes 

a  = 


We  have  here,  as  before,  ^  =  fil  —  o>  ;  and  the  value  of  pt  is 
also  given  in  the  Ephemeris  for  the  Washington  meridian. 

If  now  for  any  assumed  time  T0  we  take  from  the  Ephemeris 
the  values  of  these  auxiliaries,  and,  after  computing  a,  6,  and  c 
by  (560),  find  that  a  differs  from  j/6c,  the  assumed  time  requires 
to  be  corrected;  and  the  correction  is  found  by  the  following 


m  = 
a*,  V,  m'  =  the  changes  of  a,  6,  m,  in  one  second, 

r  =  the  required  correction  of  the  assumed  time ; 

then  at  the  time  of  beginning  or  ending  of  the  eclipse  we  must 
have 

a  +  afr  =  m  +  m'r 

whence 

m  —  a 


To  find  a'  we  have,  by  differentiating  the  value  of  a  and  de- 
noting the  derivatives  by  accents, 

a'  =  A'  —  ft'f  cos  j/  cos  »3 

Vou  L— 33 


514  SOLAR    ECLIPSES. 

in  which  p.'  denotes  the  change  of  pl  in  one  second,  and  is  the 
same  as  the  pf  of  our  former  method  divided  by  3600. 

To  find  mf  we  have,  following  the  same  notation,  and  neglect- 
ing the  small  changes  of  E,  F,  6r,  H,  £,  and  /, 

B'=       y'=Cf 

V   =       Br  —  p!  G  p  cos  <?'  sin  # 

cf   =  —  C'  -\-  fji'  H  p  cos  <p'  sin  & 

Since  /  is  small,  we  may  in  these  approximate  expressions  put 
Gr  =  H,  and  hence 

V  =  —  d  =  B'  —  p!  G  p  cos  <p'  sin  *  (561*) 

Now,  from  the  formula  m2  —  be,  we  derive 

2  mm'  ==  cV  +  bc'  =  (c  —  6)  V 


which,  if  we  assume 


becomes 

m'=  —  bf  cot  Q 

and  therefore  r  is  found  by  the  formula 

m  —  a 


a'  -f-  b'  cot  Q 


(563) 


The  Ephemeris  gives  also  the  values  of  A',  Bf,  and  C",  which 
are  the  changes  of  A,  B,  and  C  in  one  second.  These  changes 
being  very  small,  the  unit  adopted  in  expressing  them  is  .000001  ; 
so  that  the  above  value  of  r,  as  also  the  value  of  p!  in  (561), 
must  be  multiplied  by  106.  The  formulae  (560-563)  then  agree 
with  those  given  in  the  explanation  appended  to  the  Ephemeris. 

It  is  easily  seen  that  Q  here  denotes  the  same  angle  as  in  the 
preceding  articles  ;  for  we  have  at  the  instant  of  contact 

b'          2m        x  —  Z 
tan  Q  =  --  r  = 
' 


m'        b  —  c       y  —  i) 

Examples  of  the  application  of  this  method  are  given  in  every 
volume  of  the  American  Ephemeris. 


CORRECTION    FOR    REFRACTION. 


516 


326.  The  preceding  articles  embrace  all  that  is  important  in 
relation  to  the  prediction  of  solar  eclipses.     Since  absolute  rigor 
is  not  required  in  mere  predictions,  I  have  thus  far  said  nothing 
of  the  effect  of  refraction,  which,  though  extremely  small,  must 
be  treated  of  before  we  proceed  to  the  application  of  observed 
eclipses,  where  the  greatest  possible  degree  of  precision  is  to  be 
sought. 

CORRECTION    FOR    ATMOSPHERIC    REFRACTION    IN    ECLIPSES. 

327.  That  the  refraction  varies  for  bodies  at  different  distances 
from  the  earth  has  already  been  noticed  in  Art.  106 ;  but  the 
difference  is  so  small  that  it  is  disregarded  in  all  problems  in 
which  the   absolute   position  of  a   single   body  is  considered. 
Here,  however,  where  two  points  at  very  different  distances  from 
the  earth  are  observed  in  apparent  contact,  it  is  worth  while  to 
inquire  how  far  the  difference  in  question  may  affect  our  results. 

Let  8MDA,  Fig.  44,  be  the  path 
of  the  ray  of  light  from  the  sun's 
limb  to  the  observer  at  A,  which 
touches  the  moon's  limb  at  M ;  8MB 
the  straight  line  which  coincides  with 
this  path  between  S  and  M,  but  when 
produced  intersects  the  vertical  line 
of  the  observer  in  B.  It  is  evident 
that  the  observer  at  A  sees  an  ap- 
parent contact  of  the  limbs  at  the 
instant  when  an  observer  at  B  would 
see  a  true  contact  if  there  were  no 
refraction.  Hence,  if  we  substitute 
the  point  B  for  the  point  A  in  the 
formula  of  the  eclipse,  we  shall  fully  take  into  account  the  effect 
of  refraction. 

For  the  purpose  of  determining  the  position  of  the  point  B, 
whose  distance  from  A  is  very  small,  it  will  suffice  to  regard  the 
earth  as  a  sphere  with  the  radius  p  =  CA.  It  is  one  of  the  pro- 
perties of  the  path  of  a  ray  of  light  in  the  atmosphere  that  the 
product  qtj.  sin  i  is  constant  (Art.  108),  q  denoting  the  normal  to 
any  infinitesimal  stratum  of  the  atmosphere  at  the  point  in  which 
the  ray  intersects  the  stratum,  fj.  the  index  of  refraction  of  that 
stratum,  and  i  the  angle  which  the  ray  makes  with  the  normal. 


516  SOLAR    ECLIPSES. 

If,  then,  />,  JHO,  Z'  denote  the  values  of  q,  p,  and  %  for  the  point  A, 
we  have,  as  in  the  equation  (149), 

qrj.  sin  z  =  pfj.Q  sin  ^' 

in  which  Z'  is  the  apparent  zenith  distance  of  the  point  Jf,  and 
//0  is  the  index  of  refraction  of  the  air  at  the  observer. 

Now,  let  us  consider  the  normal  q  to  be  drawn  to  a  point  D  of 
the  ray  where  the  refractive  power  of  the  air  is  zero,  that  is,  to 
a  point  in  the  rectilinear  portion  of  the  path  where  //  —  1.  Then 
our  equation  becomes 

q  sin  i  =  p,v.0  sin  Z' 

in  which  q  =  CD,  i  =  MDF  =  CDB.  Putting  Z  =*  the  true 
zenith  distance  of  M  =  MBV,  and  s  =  the  height  of  B  above 
the  surface  of  the  earth  =  AB,  the  triangle  CDB  gives 


which  with  the  preceding  equation  gives 

• 


In  order  to  substitute  the  point  B  for  the  point  A  in  our  com- 
putation of  an  eclipse,  we  have  only  to  write  p  -\-  s  for  p  in  the 

equations  (483),  or  pt  1  -+   -  jfor  p.     Therefore,  when  we  have 

computed  the  values  of  log  £,  log  iy,  and  log  £  by  those  equa- 
tions in  their  present  form,  we  shall  merely  have  to  correct  them 

by  adding  to  each  the  value  of  log  I  1  -f    -  I.     This  logarithm 

may  be  computed  by  (564)  for  a  mean  value  of  ^  (==  1.0002800) 
and  for  given  values  of  Z.  For  Z  we  may  take  the  true  zenith 
distance  of  the  point  Z  (Art.  289),  determined  by  a  and  d.  But 
by  the  last  equation  of  (483)  we  have  so  nearly  cos  Z  =  £  that 
in  the  table  computed  by  (564)  we  may  make  log  f  the  argu- 
ment, as  in  the  following  table,  which  I  have  deduced  from  that 
of  BESSEL  (Astron.  Untersmhungen,  Vol.  II.  p.  240). 


REDUCTION  TO  THE  SEA  LEVEL. 


517 


logC 

Correction  of  logs, 
of  1,  T],  f. 

00 
9.9 
9.8 
9.7 
9.6 

0.0000000 

.0000001 
.0000002 
.0000005 
.0000008 

9.5 
9.4 
9.3 
9.2 
9.1 

0.0000014 
.0000023 
.0000035 
.0000054 
.0000081 

9.0 
8.9 

8.8 
8.7 
8.6 

0.0000119 
.0000167 
.0000225 
.0000292 
.0000367 

8.5 

0.0000446 

logC 

Correction  of  logs, 
of  £  q,  C- 

8.5 
8.4 
8.3 
8.2 
8.1 

0.0000446 
.0000525 
.0000602 
.0000672 
.0000734 

8.0 
7.9 

7.8 
7.7 
7.6 

0.0000788 
.0000835 
.0000875 
.0000909 
.0000937 

7.4 
7.2 
7.0 

6.5 
6.0 

0.0000978 
.0001006 
.0001023 
.0001044 
.0001051 

—  oo 

0.0001054 

The  numbers  in  this  table  correspond  to  that  state  of  the  at- 
mosphere for  which  the  refraction  table  (Table  II.)  is  computed; 
that  is,  for  the  case  in  which  the  factors  ft  and  f  of  that  table  are 
each  —  1.  For  any  other  case  the  tabular  logarithm  is  to  be 
varied  in  proportion  to  ft  and  f. 

It  is  evident  from  this  table  that  the  effect  of  refraction  will 
mostly  be  very  small,  for  so  long  as  the  zenith  distance  of  the 
moon  is  less  than  70°  we  have  log  £  >  9.53,  and  the  tabular 
correction  less  than  .000001.  From  the  zenith  distance  70°  to 
90°  the  correction  increases  rapidly,  and  should  not  be  neglected. 

CORRECTION  FOR  THE  HEIGHT  OF  THE  OBSERVER  ABOVE  THE 
LEVEL  OF  THE  SEA. 

328.  If  sr  is  the  height  of  the  observer  above  the  level  of  the 
sea,  it  is  only  necessary  to  put  p  -f  s'  for  p  in  the  general  formulse 
of  the  eclipse  ;  and  this  will  be  accomplished  by  adding  to  log  c, 

log  7,  and  log  £  the  value  of  log  (  1  +  ~  1,  which  is  (  M  being 
the  modulus  of  common  logarithms) 


But  s'  is  always  so  small  in  comparison  with  p  that  we  may 


518  SOLAR    ECLIPSES. 

neglect  all  but  the  first  term  of  this  formula ;  and  hence,  by 
taking  a  mean  value  of  />  (for  latitude  45°)  and  supposing  s'  to 
be  expressed  in  English  feet,  we  find 

Correction  of  log  £,  log  y,  log  C  =  0.00000002079  s'       (565) 

For  example,  if  the  point  of  observation  is  1000  feet  above 
the  level  of  the  sea,  we  must  increase  the  logarithms  of  £,  57, 
and  £  by  0.0000208. 

If  s'  is  expressed  in  metres,  the  correction  becomes  0.000000064  s'. 

APPLICATION  OF  OBSERVED  ECLIPSES  TO  THE  DETERMINATION  OF  TER- 
RESTRIAL LONGITUDES  AND  THE  CORRECTION  OF  THE  ELEMENTS 
OF  THE  COMPUTATION. 

.  829.  To  find  the  longitude  of  a  place  from  the  observation  of  an 
eclipse  of  the  sun. — The  observation  gives  simply  the  local  times 
of  the  contacts  of  the  discs  of  the  sun  and  moon :  in  the  case  of 
partial  eclipse,  two  exterior  contacts  only ;  in  the  case  of  total  or 
annular  eclipse,  also  two  interior  contacts. 

Let 

w  =  the  west  longitude  of  the  place, 

t  =  the  local  mean  time  of  an  observed  contact, 

/j.  =  the  corresponding  local  sidereal  time. 

The  conversion  of  t  into  //  requires  an  approximate  knowledge 
of  the  longitude,  which  we  may  always  suppose  the  observer  to 
possess,  at  least  with  sufficient  precision  for  this  purpose. 

Let  T0  be  the  adopted  epoch  from  which  the  values  of  x  and  y 
are  computed  (Art.  296),  and  let 

x0,  y0  =  the  values  of  x  and  y  at  the  time  T0, 

x',  y'  =  their  mean  hourly  changes  for  the  time  t  -{-  to; 

then,  if  we  also  put 

T  =  t  +  o>  —  TQ  (566) 

the  values  of  x  and  y  at  the  time  t  -\-(o  (which  is  the  time  at  the 
first  meridian  when  the  contact  was  observed)  are 

*<>  -f  x'r>  y*  +  y'r 

The  values  of  x'  and  y'  to  be  employed  in  these  expressions 
may  be  taken  for  the  time  t  +  <*>  obtained  by  employing  the 


LONGITUDE.  519 

approximate  value  of  o>,  and  will  be  sufficiently  precise  unless 
the  longitude  is  very  greatly  in  error. 

The  quantities  I  and  i  change  so  slowly  that  their  values 
taken  for  the  approximate  time  t  -f  a>  will  not  differ  sensibly 
from  the  true  ones.  For  the  same  reason,  the  quantities  a  and  d 
taken  for  this  time  will  be  sufficiently  precise :  so  that,  the  latitude 
being  given,  the  co-ordinates  £,  ^,  f  of  the  place  of  observation 
may  be  correctly  found  by  the  formulae  (483).  Since,  then,  at 
the  instant  of  contact  the  equation  (490)  or  (491)  must  be  exactly 
satisfied,  we  have,  putting  L  =  I  —  if, 

'  56T 

in  which  r  is  the  only  unknown  quantity.     Let  the  auxiliaries 
w,  M,  n,  Nl>Q  determined  by  the  equations 


m  sin  M  =  x0  —  £  n  sin  N  =  x' 

m  cos  M  =  y0  —  77  n  cos  N  =  y' 


}    (568) 


then,  from  the  equations 


L  sin  Q  =  m  sin  M  -\-  n  sin  N  .  r 
L  cos  Q  =  m  cos  M  -\-  n  cos  N  .  r 


by  putting  <\J/  =  Q  —  N,  we  obtain 

m  sin  (M  —  « 


sin      = 


_  L  cos  4       m  cos  ( Jf  —  JV) 
7»  n 


m  sin  (M  —  N  —  4) 
n  sin  4 


(569) 


where  the  second  form  for  r  will  be  the  more  convenient  except 
when  sin  ^  is  very  small.  As  in  the  similar  formulse  (553),  the 
angle  ^  must  be  so  taken  that  L  cos  ^  shall  be  negative  for 
first  contacts  and  positive  for  last  contacts,  remembering  that  in 
the  case  of  total  eclipse  L  is  a  negative  quantity. 

Having  found  r,  the  longitude  becomes  known  by  (566),  which 

gives 

*  =  T0  -  t  +  r  (570) 


520  SOLAR    ECLIPSES. 

If  the  observed  local  time  is  sidereal,  let  /*0  be  the  sidereal 
time  at  the  first  meridian,  corresponding  to  T0;  then,  r  being 
reduced  to  sidereal  seconds,  we  shall  have 

w  =  HQ  —  /j.  -f  r 

and  this  process  will  be  free  from  the  theoretical  inaccuracy 
arising  from  employing  an  approximate  longitude  in  converting 
fj.  into  t. 

The  unit  of  T  in  (569)  is  one  mean  hour ;  but,  if  we  wrrite 

h  L  cos  4,        Jim  cos (M  —  N) 


m   sin  (M  —  N  —  40 
n  sin  4, 

•we  shall  find  T  in  mean  or  sidereal  seconds,  according  as  we  take 
h  =  3600,  or  h  =  3609.856. 

330.  The  rule  given  in  the  preceding  article  for  determining 
the  sign  of  cos  ^  (which  is  that  usually  given  by  writers  on  this 
subject)  is  not  without  exception  in  theory,  although  in  practice 
it  will  be  applicable  in  all  cases  where  the  observations  are 
suitable  for  finding  the  longitude  with  precision  ;  and,  were  an 
exceptional  case  to  occur  in  practice,  a  knowledge  of  the  approxi- 
mate longitude  would  remove  all  doubt  as  to  the  sign  of  the  term 


.     But  it  is  is  easy  to  deduce  the  mathematical  condition 
n 

for  this  case. 

At  the  instant  of  contact,  the  quantity 

(.r0-£-f  *'r)'-f  (^-,+yr)* 

is  equal  to  L2.  At  the  next  following  instant,  when  r  becomes 
T  -\-  ch,  it  is  less  or  greater  than  L2  according  as  the  eclipse  is 
beginning  or  ending.  If  then  we  regard  L2  as  sensibly  constant, 
the  differential  coefficient  of  this  quantity  relatively  to  the  time 
must  be  negative  for  first  and  positive  for  last  contacts.  The 
half  of  this  coefiicient  is 

(x9  -  ?  -f  a^)  <x  -  o  +  G/O  -  7  +  /*•)  <y  -  v) 

(where  the  derivatives  of  £  and  r]  are  denoted  by  £'  and  57'),  or,  by 
(567),  putting  N  +  ^  for  §, 

L  [sin  (N  -f  4)  (x1—  £')  +  cos  (N  +  40  O/'  —  V)] 


LONGITUDE.  521 

Computing  £'  and  tf  by  the  formulae  (551),  or,  in  this  case,  by 
£'=  fi'p  cos  <?'  cos  (fJL  —  a)  V==  /*'£  sm  d 

and  putting 

n'  sin  N'  =  x'  —  £'  rc'  cos  N'  =  y'  —  if 

the  above  expression  becomes 


Hence,  when  L  is  positive,  that  is,  for  exterior  contacts  and 
interior  contacts  in  annular  eclipse,  ^  must  be  so  taken  that 
cos(N—  N'  +  $)  shall  be  negative  for  first  and  positive  for  last 
contact.  That  is,  for  first  contact  fy  must  be  taken  between 
Nf—  N+  90°  andN'—  N+  270°;  and  for  last  contact  between 
N'—N  -}-  90°  and  N'—N—  90°.  For  total  eclipse,  invert  these 
conditions. 

In  Art.  322,  we  have  N  =  Nf,  and  hence  the  rule  given  for 
the  case  there  considered  is  always  correct. 

331.   To  investigate  the  correction  of  the  longitude  found  from  an 
observed  solar  eclipse,  for  errors  in  the  elements  of  the  computation. 
Let 
AX,  Ay,  &L  =  the  corrections  of  x,  y,  and  L,  respectively, 

for  errors  of  the  Ephemeris, 

A£,  A>?  —  the  corrections  of  £  and  ^  for  errors  in  p  and  ^>', 
Ar  =  the  resulting  correction  of  r. 

The  relation  between  these  corrections,  supposing  them  very 
small,  will  be  obtained  by  differentiating  the  values  of  L  sin  Q 
and  L  cos  Q  of  the  preceding  article,  by  which  we  obtain 

&L  sin  Q  -f  L  COS§A§  =  AJC  —  A£  -f-  X'AT 
AJD  cos  Q  —  L  sin  Q  A  Q  =  AZ/  —  Ai?  -f-  y'  Ar 

where  AX  and  A^,  being  taken  to  denote  the  corrections  of 
x  =  XQ-\-  x'r  and  y  =  yQ-\-  y'-c,  include  the  corrections  of  x'  and  y'. 
Substituting  in  these  equations  n  sin  N  for  x'  and  n  cos  N  for 
y',  and  eliminating  A§,  we  find 

A  L  =  (AX  —  A£)  sin  (g  -f  (A#  —  AIJ)  cos  g  +  n  cos  (Q  —  JV)  .  Ar 
and  substituting  for  Q  its  value  N+  ^ 


.  sin  (A -f- 4)  ,cos,.,    i    ^, 

AT  —  —  (AX  —  A£)  -  -  (Ay  —  Ary)  -  ^LL  _|_ 

n  .cos  4  n  cos  4-  n  cos  4 


522  SOLAR    ECLIPSES. 

i 

or 

AT  =r^  —   —  (AX  sin  JV-f-  A?/  cos  Ar)  -|  —  (  —  A.r  cos  N-\-  Ay  sin  JV)  tan 

-|  --  (A£  sin  JV-f  AT?  cos  JV)  --  (—  A*  cosJV-f-  A>?  sin  W)  tan 


71 

which  is  at  once  the  correction  of  r  and  of  the  longitude,  since 
we  have,  by  (570),  AW  =  AT. 

332.  In  this  expression  for  AT,  the  corrections  AX,  A,?/,  &c.  have 
particular  values  belonging  to  the  given  instant  of  observation 
or  to  the  given  place.  In  order  to  render  it  available  for  deter- 
mining the  corrections  of  the  original  elements  of  computation, 
we  must  endeavor  to  reduce  it  to  a  function  of  quantities  which 
are  constant  during  the  whole  eclipse  and  independent  of  the 
place  of  observation.  For  this  purpose,  let  us  first  consider 
those  parts  of  AT-  which  involve  AX  and  &y.  For  any  time  Tv  at 
the  first  meridian,  we  have 

x  =  x0  +  n  sin 
y  =  iJo+nGos 
whence 

x  sin  N  -f  y  cos  N=       XQ  sin  N  -f  y0  cos  N-{-  n(^—  TQ) 
-  x  cos  N  +  y  sin  N=  —  x0  cos  N  -\-  y0  sin  N 

The  last  of  these  expressions,  being  independent  of  the  time,  is 
constant.     If  we  denote  it  by  x  ;  that  is,  put 

x  =  —  XQ  cos  N  -f  y0  sin  N  =  —  x  cos  N  +  #  sin  JV"      (572) 
we  shall  find  from  the  two  expressions 

xx  +  yy  -  xx  +  Dr0sin  ^+  y0cos  ^+  n  (T±-  T0)]2     (573) 


This  equation  shows  that  the  quantity  \/xx  -f  yy,  which  is  the 
distance  of  the  axis  of  the  shadow  from  the  centre  of  the  earth, 
can  never  be  less  than  the  constant  x,  and  it  attains  this  minimum 
value  when  the  second  term  vanishes,  that  is,  when 


and  hence  when 

T,  =  T0  -  1  (x0  sin  N  +  y.  cos  N}  (574) 


LONGITUDE.  523 

which  formula,  therefore,  gives  the  time  Tt  of  nearest  approach  of 
the  axis  of  the  shadow  to  the  centre  of  the  earth,  while  (572) 
gives  the  value  of  the  distance  of  the  axis  from  the  centre  of  the 
earth  at  this  time.  By  the  introduction  of  the  auxiliary  quanti- 
ties Tv  and  x,  we  can  express  the  corrections  involving  AX  and  &y 
in  their  simplest  form  ;  for  we  have  now,  for  the  time  of  obser- 

vation t  -f-  OJj 

x  sin  N  -\-  y  cos  N  =  xn  sin  N  -f-  yQ  cos  N  -{-  n(t  -\-  to  —  T0) 
=  n  (t  -f  «,  -  T,) 

and  if  A7^,  AT1,  and  A#  are  the  corrections  of  ft,  Tv  and  x  on 
account  of  errors  in  the  elements,  we  have 

AJ;  sin  N  -f  A  y  cos  N  =  —  n  A  Tl  -f  (f  +  «>  —  7^)  AW     ) 
-  A£  cos  ^V  -f  AZ/  sin  JV  =  AX  j     '   '   ' 

These  expressions  reduce  those  parts  of  Ar  which  involve  AX  and 
AI/  to  functions  of  A  T^  A??,  and  A^,  which  may  he  regarded  as 
constant  quantities  for  the  same  eclipse. 

We  proceed  to  consider  those  parts  of  Ar  which  involve  A£ 
and  A^.  These  corrections  we  shall  regard  as  depending  only 
upon  the  correction  of  the  eccentricity  of  the  terrestrial  meridian; 
for  the  latitude  itself  may  always  be  supposed  to  be  correct, 
since  it  is  easily  obtained  with  all  the  precision  required  for  the 
calculation  of  an  eclipse  ;  the  values  of  a  and  d  depend  chiefly 
on  the  sun's  place,  which  we  assume  to  be  correctly  given  in  the 
Ephemeris  ;  and  fj.  is  derived  directly  from  observation. 
we  have  (Art.  82),  e  being  the  eccentricity  of  the  meridian, 

cos  <p  (\  —  ee)  sin  <p 

p  cos  <p'  =  -  f)  sin  f  =  — 

' 


whence,  by  differentiation, 

A  .  p  cos  (f>f  op  sin2 

' 


Aee  2(1  — ee)2 

pp  sin2  <p'         p  sin 


_ 

1 


:  (^i  —  ee;         i  —  ee 
or,  putting 

p  sin  tp' 


1  —  ee 


524  SOLAR    ECLIPSES. 

A  .  p  COS  <f>' 

—  =   £  pftp   COS   <f 


.  p  sin  <p  .       . 

— —  —  2  fti3  p  sin  <p  — 


From  the  values 

£  =  p  cos  <f  sin  Qt  —  a) 

ij  =  p  sin  <p'  cos  d  —  p  cos  <p'  sin  d  cos  (^  —  a) 

we  deduce 

A* 


and  hence 


(576) 


cos 

Aee  Aee 


A£  sin  JV  -}-  A?  cos  N  =  %(3(3  (      £  sin  JV  -(-  ^  cos  JV)  Aee  —  /?  cos  d  cos  ^V  Aee 
—  A*  cos  JV  -f  A?  sin  N  =  $J3J3  (  —  f  cos  N  +  if  sin  JV)  A«e  —  /9  cos  rf  sin  7^  bee 

The  values  of  £  and  37  may  be  put  under  the  forms 
$  =  XQ—  (XQ  —$*)  =  x0—m  sin  M 

y  =  y«  —  Q/o  —  ^)  =  y0  —  m  cos  ^ 

by  which  the  second  members  of  the  preceding  expressions  are 
changed  respectively  into 


£  /?/?  [      x0  sin  JV  -f-  y0cos  N  —  m  cos  (J/  —  N)]  Aee  —  /?  cos  d  cos  ^V  Aee 
and      ^  /3/3  [—  a;0cos^V  -f  y0sin  N  +  m  sin  (M  —  N)]  Aee  —  p  cos  d  sin  JV  Aee 

or,  by  (574)  and  (572),  into 

£  (3(3  [n  (7^  —  7\)  —  TO  cos  (J/ —  N)~\  Aee  —  /3  cos  d  cos  jV  Aee 
and  £  /?#  [  x  -f-  m  sin  (J/  —  JV)]  Aee  —  (3  cos  d  sin  JV  A?e 

or,  again,  by  (569)  and  (570),  into 

» 

J  /?/?[»  (<  +  w  —  7\)  —  L  cos  -^l  ^f«  —  (3  cos  e?  cos  N  Aee 
and  £  /5/3  f  x  4-  L  sin  ^l  ^ee  —  ft  cos 


.Hence,  that  part  of  Ar  which  depends  upon  Aee  is  equal  to 

g_  [n  (t  -|-  w  _  7^)  —  x  tan  $  —  L  sec  4,]  Aee  - 


When  these  substitutions  are  made  in  (571),  we  have 

Ar  =  Aw  =  h  A  7^  -f-  h  tan  4, .  ^  —  A  (t  +  u  —  7\)  —  -f  A  sec  4. 

n  n  n 

-  r*fl*  [»  (<  +  w  -  ^i)  -  *  tan  *  -  L  sec  4]  -  0co8rfcM  ^+ *>]  Aee  (577 
n  \_  cos  4/  J 


LONGITUDE.  525 

where  we  have  multiplied  by  h  to  reduce  to  seconds.  The  unit 
is  either  one  second  of  mean  or  one  second  of  sidereal  time, 
according  as  r  is  in  mean  or  sidereal  time.  If  the  former,  we 
take  h  =  3600;  if  the  latter,  h  =  3610. 

333.  The  transformations  of  the  preceding  article  have  led  us 
to  an  expression  in  which  the  corrections  A  Tv  AX,  An,  and  &ee  are 
all  constants  for  the  earth  generally,  and  which,  therefore,  have 
the  same  values  in  all  the  equations  of  condition  formed  from 
the  observations  in  various  places.  But  a  still  further  transform- 
ation is  necessary  if  we  wish  the  equation  to  express  the  rela- 
tion between  the  longitude  and  the  corrections  of  the  Ephemeris, 
so  that  we  may  finally  be  enabled  not  only  to  correct  the  longi- 
tudes, but  also  the  Ephemeris. 

Since  A  Tv  AX,  AH  are  constant  for  the  whole  eclipse,  we  can 
determine  them  for  any  assumed  time,  as  the  time  JJ  itself.  For 
this  time  we  have 


(578) 


x  sin  N  -f-     y  cos  N  =  0 
—  x  cos  N  -j-     ?/  sin  N  =  x 

A.r  sin  JV  -f  Ay  cos  JV  —  —  n  A  7^ 
—  AX  cos  N  -\-  &y  sin  N  =  AX 

The  general  values  of  x  and  #  (482)  may  be  thus  expressed: 

y  ~v 

x  =  -. — -  y  =  — — 

sm  TT  sin  TT 

where 

X  =  cos  d  sin  (a  —  a)  Y  =  sin  d  cos  d  —  cos  8  sin  d  cos  (a  —  a) 

From  these  we  deduce 


ATT  A  Y  A?r 


x  -  Ay  =  — v 

-friY-k    M  J  oi«.     _  •? 


sin  ^  tan  TT  sin  TT        *  tan  x 


whence 


AX  sin  N  -4-  A  Y  cos  N       ,  ,rx     ATT 

Az  sm  N  -f-  Ay  cos  N  = -31 (x  sm  N  -f  y  cos 


tan  TT 

ATT 


»r  i  •      nr       —  ,  ,r  .     ™ 

—  Ar  cos  #  -f  Ay  sin  JV  =  —  —  (-  (x  cos  JV^  —  v  sm  N) 

sin  TT  tan  TT 

and  for  the  time  7\  these  become,  according  to  (578), 

AFcos  JV 


„, 

—  n  A  J:    ^=r: 


sm 


AYsin  JV  ATT 

X  -- 


sin  r  tanr 


526  SOLAR   ECLIPSES. 

Again,  by  differentiating  the  values  of  X  and  Y,  we  have 


—  cos  3  cos  (a  —  a)  A(a  —  a)  —  sin  3  sin  (a  —  a)  &3 
A  Y  =  [cos  3  cos  d  -f-  sin  3  sin  d  cos  (a  —  a)]  A£ 
-  [sin  3  sin  6?  -f-  cos  3  cos  c?  cos  (a  —  a)]  Ad 
-j-    cos  3  sin  d  sin  (a  —  a)  A(a  —  a] 

But  for  the  time  of  nearest  approach  we  may  take  a  =  a  and 
put  cos(^  —  d)  =  1,  whence 

A  JT  =  cos  3  .  A(a  —  a)  A  F  =  A(<5  —  <f) 

so  that 


sin  N  cos  3  .  A(a  —  a)  -|-  cos  N  .  A(<5  —  d) 

sin  n 

—  cos^V  cos  3.  A(a — a)  -f-  sin  JV.  A (5  —  d)  ATT 


sin  TT  tan  - 


(579) 


To  find  An,  which  depends  upon  the  corrections  of  x'  and  ?/', 
we  observe  that  xr  and  ,?/',  regarded  as  derivatives  of  x  and  j/,  are 
of  the  form 

i  <*r    i 


_        ___ 

dT  sinr  cZr'sin;r 

But  -^=-  and  -—  -  depend  upon  the  changes  of  the  moon's  right 

(JL  JL  CL  JL 

ascension  and  declination,  which  for  the  brief  duration  of  an 
eclipse  are  correctly  given  in  the  Ephemeris.  The  errors  of  x' 
and  y',  therefore,  depend  upon  those  of  TT  :  so  that  if  we  write 

a  ,         b 

x'  =  —  yr  =  -^ 

sinr  sin?:  ^ 

and  regard  a  and  6  as  correct,  we  find 

,    A:r  ,    Ar 

AX'  =  —  x'  -  A  y  '  =  —  y  - 

tan  TT  tan  - 

• 
From  the  equations  n  sin  -ZV=  x',  n  cos  N=  y',  we  have 

AW  sin  N  -\-  n  A^V  cos  N  =  &xr  =  —  n  sin  JV  •  - 

tanrr 

An  cos  N—  n  A^sin  N  —  AV'=  —  ncosJV  --  — 

tanr 


LONGITUDE.  527 

whence,  by  eliminating  A-ZV,* 

^  =  _  -^L_  (580) 

n  tan  TT 

Since  A(CC  —  a),  A(<?  —  rf),  ATT  will  in  practice  be  expressed  in 
seconds  of  arc,  we  should  substitute  for  them  A  (a  —  a)  sin  1", 
A(#  —  d)  sin  1",  ATI  sin  \"  in  the  above  expressions  ;  but  if  we  at 
the  same  time  put  it  sin  V  for  sin  n  and  tan  n,  the  factor  sin  V 
will  disappear. 

To  develop  A£,  we  may  neglect  the  error  of  the  small  term  i£ 
and  assume  A£  =  A/.  We  have  from  (486)  and  (488),  by 

neglecting  the  small  term  k  sin  ;TO  and  putting  g  =  1,  z  =  —  -  , 
the  following  approximate  expression  for  I: 


r'  sin  TT 
which  gives 

A-ff  H     ATT 

A£  =  £d  =  --  ±  A^  --  ---  -  (581  ) 

r';r  r'*     ^ 

Substituting  the  values  of  A  JJ,  A*,  ATI,  and  A£  given  by  (579), 
(580),  and  (581),  in  (577),  and  putting 


the  formula  becomes,  finally, 

Au  =  —  v  [      sin  N  cos  J.A(a  —  a)  -|-  cos  JV.A((?  —  «?)] 

-|-  v  [  —  cos  -A7"  cos  d.A(a  —  a)  -f-  siQ  N.£^(S  —  c?)]  tan 


\n  (t  +  w  —  7\)  —  *  tan  4.  --  r  sec  4,  j 


cos  4* 

(682) 

where  the  negative  sign  of  TTA&  is  to  be  used  for  interior  contacts. 
It  is  easily  seen  that  TTA&  represents  very  nearly  the  correction 


*  The  angle  N  is  independent  of  errors  in  TT,  since  tan  N  =  - :  so  that  we  might 
have  taken  A^=0. 


528  SOLAR    ECLIPSES. 

A  TT 
of  the  moon's  apparent  semidiameter,  and  — **  that  of  the  sun's 

semidiameter ;  and  that  x&ee  is  the  correction  of  the  assumed 
reduction  of  the  parallax  for  the  latitude  90°. 

334.  Discussion  of  the  equations  of  condition  for  the  correction  of 
the  longitude  and  of  the  elements  of  the  computation. — The  longitude 
at  found  by  the  equation  (570),  (Art.  329),  requires  the  correction 
AOJ  of  (582).  If,  for  brevity,  we  put 

Y=       sin  N  cos  d  A  (a  —  a)  -f-  cos  N&(d  —  d)    \     /-COON 
$  =  —  cos  N  cos  d  A(a  —  a)  -f  sin  N &(d  —  d)     ) 
and 

o/=  the  true  longitude, 

we  have  the  equation  of  condition 

a/  =  «»  -f-  Aa>  =  it)  —  \>f  -\-  \>  tan  4- .  #  -f  &c.  (584) 

If  the  eclipse  has  been  observed  at  several  places,  we  can  form 
as  many  such  equations  as  there  are  contacts  observed.  If  the 
observations  are  complete  at  all  the  places,  we  can,  for  the  most 
part,  eliminate  from  these  equations  the  unknown  corrections  of 
the  elements,  and  determine  the  relative  longitudes  of  the  several 
places ;  and  if  the  absolute  longitude  of  one  of  the  places  is 
known,  that  of  each  place  will  also  be  determined. 

I  shall  at  first  consider  only  the  terms  involving  f  and  #.  The 
quantity  vy  is  a  constant  for  all  the  places  of  observation,  and 
combines  with  a;,  so  that  it  cannot  be  determined  unless  the 
longitude  of  at  least  one  of  the  places  is  known.  If  then  we  put 

Q  =  ot'  -}-  vf  a  —  v  tan  ^ 

the  equations  of  condition  will  assume  the  form 

Q  —  a&  —  a>  —  0 

Suppose,  for  the  sake  of  completeness,  that  the  four  contacts 
of  a  total  or  annular  eclipse  have  been  observed  at  any  one  place, 
and  that  the  values  of  the  longitude  found  from  the  several  con- 
tacts by  Art.  329  are  w^  w2,  w31  w4.  We  then  have  the  four  equa- 
tions 

[1]    Q  —  flj  &  —  wl  —  0 

[2]    Q  —  a3  $  —  a>2  =  0 

[3]     Q  —  aA  #  —  a>3  =  0 

[4]     Q  —  a4  *  —  o;4  =  0 


LONGITUDE.  529 

where  the  numerals  may  be  assumed  to  express  the  order  in 
which  the  contacts  are  observed  ;  [1]  and  [4]  being  exterior,  and 
[2]  and  [3]  interior.  In  a  partial  eclipse  we  should  have  but  the 
1st  and  4th  of  these  equations. 

Since  exterior  contacts  cannot  (in  most  cases)  be  observed  with 
as  much  precision  as  interior  ones,  let  us  assign  different  weights 
to  the  observations,  and  denote  them  by  pv  p2,  p^p^  respectively. 
Combining  the  four  equations  according  to  the  method  of  least 
squares,  we  form  the  two  normal  equations 

IP  ]fl-[>  ]*-[>  ]=0 
[pa]  Q  —  [paa]  &  —  [paio]  =  0 

where  the  rectangular  brackets  are  used  as  symbols  of  summa- 
tion. From  these,  by  eliminating  J2,  and  putting 


we  find 

PV  +  Q  =  0  (585) 

from  which  the  value  of  &  would  be  determined  with  the  weight 
P.  But  the  computation  of  Q  under  this  form  is  inconvenient. 
By  developing  the  quantities  P  and  $,  observing  that  \_paa]  = 

Piai  +  Piat  +  Ws2  +  P*a*9  &c->  we  sna 


P  _  PI  Pi  (ai  —  <*2)2  +  PI  P3  (a\  —  "a)2  +  Pi  i>4  (ai  —  ««)* 

/>!+  ^2+^3+^4 

I  Pt  PZ  K  —  "3)2  +  PI  P*  (a-z  —  a*Y  +  r»  p*  (ff»  —  q4)a 

Pl  +  Pl  +  ^3  +  1>4 

Q  _P\  Pi  (a\  —  q2)  ("i  —  S)  +  Pi  Ps  (ai  —  qa)  ("i—  S)  +  Pi  P*  (a\  —  "4)  K  —  "4) 

Pi  +  Pt  +  P*  +  P\ 

—  "3)  K  —  "a)  +  P*  PA.  (ff2  —  a4)  K  —  6)4)  +  ^3^4  K  —  °4)  ("a  —  "4) 


. 

These  forms  show  that  if  we  subtract  each  of  the  equations  [1], 
[2],  [3]  from  each  of  those  that  follow  it  in  the  group,  whereby 
we  obtain  the  six  equations 

(a,  —  «,)  *  +  «t  —  w^  =  0 


VOL.  I. 


530  SOLAR    ECLIPSES. 

and  combine  these  six  equations  according  to  the  method  of 
least  squares,  taking  their  weights  to  be  respectively 


Pi  P*  Pi  P 


r* &c. 


PI  +  p»  t 

we  shall  arrive  at  the  same  final  equation  (585)  as  by  the  direct 
process,  with  the  advantage  of  avoiding  the  multiplication  of  the 
large  numbers  wv  co2,  &c. 

Suppose  that  at  another,  place  but  three  contacts  have  been 
observed,  the  true  longitude  being  a/',  and  the  computed  longi- 
tudes co5,  ft>6,  wr  and  that,  having  put  Q'=  w"  -f  vfr  wre  have 
formed  the  three  equations 

[5]     £'  —  a5  #  —  w5  —  0  with  the  weight  ps 
[6]     tt  -a,  *  —  «,6  =  0     "  "       p6 

[7]     Q'-a7*-a,7  =  Q     «  «       Pj 

The  subtraction  of  each  of  the  first  two  from  those  which  follow 
gives  the  three  equations 

K  —  <*6)  *  +  °>s  —  <»*  =  ° 
O5  -  «7)  *  -f  «>5  -  «>7  =  0 
(a6  —  a7)  *  +  «,.  —  w7  =  0 

of  which  the  weights  will  be  respectively,  according  to  the  above 
forms, 


^7  JP5+^6+l>7  ^5+^6  +  ^7 

and  the  combination  of  these  three  equations,  according  to 
weights,  will  give  a  normal  equation  of  the  form 

P't  -j-  §'=0 

* 

which  gives  a  value  of  #  with  the  weight  Pl  '. 

Now,  suppose  that  this  method  applied  to  all  the  observations 
at  all  the  places  has  given  us  the  series  of  equations  in  #, 

P,9  -f  Q  =  0 
P'fi  -f  Q'  =  0 
P"*+  §"=0,&c.; 

then,  since  P,  Px,  Pr',  &c.  are  the  weights  of  these  several  deter- 
minations, the  final  normal  equation  for  determining  #,  derived 
from  all  the  observations,  is 

[P]  «  +  [«]  =  0 


LONGITUDE.  531 

that  is,  it  is  simply  the  sum  of  all  the  individual  equations  in  d 
formed  for  the  places  severally. 

The  same  reasoning  is  applicable  to  any  of  the  terms  which 
follow  the  term  in  &  in  (584)  ;  so  that  if  we  suppose  all  the  terms 
to  be  retained,  this  process  gives  an  equation  in  $  for  each  place, 
in  which  besides  the  term  P&  there  will  be  terms  in  A/.:,  &H,  &c., 
and  from  all  the  equations,  by  addition,  a  final  normal  equation 
(still  called  the  equation  in  $)  as  before.  In  the  same  manner, 
final  normal  equations  in  A/J,  A/?,  &c.  will  be  formed.  Thus  we 
shall  obtain  five  normal  equations  involving  the  five  unknown 
quantities  $,  A&,  A/7,  ATT,  Ace,  which  are  then  determined  by 
solving  the  equations  in  the  usual  manner.  But,  unless  the 
eclipse  has  been  observed  at  places  widely  distant  in  longitude, 
it  will  not  be  possible  to  determine  satisfactorily  the  value  of 
A/T,  much  less  that  of  &ee.  It  will  be  advisable  to  retain  these 
terms  in  our  equations,  however,  in  order  to  show  what  effect  an 
error  in  TT  or  ee  may  produce  upon  the  resulting  longitudes. 

When  #,  &c.  have  been  found,  we  find  J2,  J2',  &c.  from  the 
equations  [1],  [2]  ____  [5],  [6]  ____  The  final  value  of  Q  will  be 
the  mean  of  its  values  [1  —  4]  taken  with  regard  to  the  weights  ; 
and  so  of  Q'  ,  &c.  Hence  we  shall  know  the  several  differences 
of  longitude 

w'  —  a>"  =  Q  —  £',  a>'  —  w"'  =  Q  —  Q",  &c. 

If  one  of  the  longitudes,  as  for  instance  w',  is  previously 
known,  we  have 


and  hence  all  the  longitudes  become  known. 

Finally,  from  the  values  of  7-  and  &  the  corrections  of  the 
Ephemeris  in  right  ascension  and  declination  are  obtained  by 
the  formulae 

cos  dA(a  —  a)  =  sin  N  .  y  —  cos  N  .  ft  j 

A(J  —  d)  =  cos  N.  r  +  sin  N  .  ft  j 

335.  When  only  two  places  of  observation  are  considered,  one 
of  which  is  known,  it  will  be  sufficiently  accurate  to  deduce  7* 
and  $  from  the  observations  at  the  known  place  (disregarding 
the  other  corrections),  and  to  employ  their  values  in  finding  the 
longitude  of  the  other  place. 


532  SOLAR    ECLIPSES. 

336.  When  good  meridian  observations  of  the  moon  are  avail- 
able, taken  near  the  time  of  the  eclipse,  the  quantities  A(OC  —  a), 
A(O  —  d)  [for  which  we  may  take  A(OC  —  a'),  &(§  —  o')],  may  be 
found  from  them.     The  terms  in  p  and  d-  may  then  be  directly 
computed  by  (583)  and  applied  to  the  computed  longitude ;  after 
which  the  discussion  of  the  equations  of  condition  may  with 
advantage  be  extended  to  the  remaining  terms. 

337.  Before  proceeding  to  give  an  example  of  the  computation 
by  the   preceding   method,  it  will  be  well  to  recapitulate  the 
necessary  formulae,  and  to  give  the    equations  of  condition  a 
practical  form. 

I.  The  general  elements  of  the  eclipse,  #,  d,  /,  log  i,  x,  ?/,  x',  ?/', 
are  supposed  to  have  been  computed  and  tabulated  as  in  Art.  297. 

II.  The  latitude  of  the  place  being  ^,  the  logarithms  of  f>  cos  <pf 
and  to  sin  <p'  are  found  by  the  aid  of  our  Table  III.,  or  by  the 
formulae  (87). 

The  mean  local  time  t  of  an  observed  contact  being  given, 
find  the  corresponding  local  sidereal  time  p. ;  also  the  time  t  -f-  to 
at  the  first  meridian,  employing  the  approximate  value  of  the 
longitude  to. 

[If  the  observed  time  is  the  sidereal  time  /*,  the  time  p  -+-  co  at 
the  first  meridian,  converted  into  mean  time,  will  give  the 
approximate  value  of  t  -j-  «.] 

For  the  time  t  +  at  take  a,  d,  I,  and  logi  from  the  eclipse 
tables,  and  compute  the  co-ordinates  of  the  place  and  the  radius 
of  the  shadow  by  the  formula 

» 

A  sin  B  =  p  sin  <p'  $  =  p  cos  <p '  sin  (p.  —  a) 

A  cos  B  =  p  cos  <p'  cos  (/Jt  —  a)  y  =  A  sin  (5  —  d) 

L  =  l  —  i:  C  =  A  cos  (S  —  d) 

When  log  £  is  small,  add  to  log  f ,  log  77,  and  log  f  the  correc- 
tion for  refraction,  from  the  table  on  p.  517. 

III.  For  the  assumed  epoch  T0  at  the  first  meridian  (being  the 
epoch  from  which  the  mean  hourly  changes  x'  and  y'  are  reck- 
oned),  take   the  values   of  x   and  y  from   the    eclipse   tables, 
denoting  them  by  XQ  and  yQ.     Also  the  mean  hourly  changes  x' 


LONGITUDE.  533 

and  yr  for  the  time  t  +  a).     Compute  the  auxiliaries  m,  M9  &c. 
by  the  formulae* 

m  sin  M  =  XQ  —  £  n  sin  N  =  x1 

m  cos  M  =  yQ  —  y  n  cos  N~  =  y' 

m  sin  (M  —  N) 
sm  4,  =  - 


where  ^  is  (in  general)  to  be  so  taken  that  L  cos  $  shall  be 
negative  for  a  first  and  positive  for  a  last  contact  (but  in  certain 
exceptional  cases  of  rare  occurrence  see  Art.  330). 

Then 

h  L  cos  4-        h  m  cos  (M  —  N) 


or,  when  sin  i^  is  not  very  small, 

hm   sinJf  —  N 


_ 


sin  4. 


If  the  local  mean  time  t  was  observed,  take  h  =  3600  in  these 
formulae,  and  then  the  (uncorrected)  longitude  is  found  by  the 
equation 

"  =  T0  -  t  +  r 

If  the  local  sidereal  time  //  was  observed,  take  h  =  3609.856, 
in  the  preceding  formulae  ;  then,  //0  being  the  sidereal  time  at  the 
first  meridian  corresponding  to  7J,  we  have 

at  =  fj.Q  —  fji  -f-  T 

The  longitudes  thus  found  will  be  the  true  ones  only  when 
all  the  elements  of  the  computation  are  correct. 

IV.  To  form  the  equations  of  condition  for  the  correction  of 
these  longitudes,  when  the  eclipse  has  been  observed  at  a  suffi- 
cient number  of  places,  compute  the  time  T^  of  nearest  approach, 
and  the  minimum  distance  #,  by  the  formulae 

T  =  T  —  —  *  8in  y      cos  ^ 


XQ  cos  N  -\-  y0  sin  N 


*  The  values  of  N  and  log  n  being  nearly  constant,  it  will  be  expedient,  where 
many  observations  are  to  be  reduced,  to  compute  them  for  the  several  integral  hours 
at  the  first  meridian,  and  to  deduce  their  values  for  any  given  time  by  simple 
interpolation. 


534  SOLAR    ECLIPSES. 

Take  /T  for  the  time  Tv  and  compute  the  -logarithm  of 

-A 

the  same  value  of  h  being  used  here  as  before. 
For  each  observation  at  each  place  compute  the  coefficients 

v  tan  ij/,  v  sec  ^,  and 

77" 

E  =  vn(t  -f  io  —  TJ  —  xv  tan  4 —  v  sec  4 

T    7T 

where  the  unit  of  t  +  co  —  T^  is  one  mean  hour, 

in  which 

H  =  959".788  log  H  =  2.98218 

ft  =P~~  log(l  — ee)  =  9: 

Then,  co'  denoting  the  true  longitude,  the  equation  of  condition  is 

A// 

r' 

where  the  negative  sign  of  the  term  y  sec  i// .  TTA/<;  is  to  be  used 
for  interior  contacts. 

The  discussion  of  the    equations  thus  formed  may  then  be 
carried   out   by  Art.  334 ;   taking   as   the   unknown  quantities 

Atf 

ft  $,  ~A/-,  — --,  ATT,  and  x&ee. 

EXAMPLE. — Find  the  longitude  of  Washington  from  the  fol- 
'  lowing  observations  of  the  solar  eclipse  of  July  28,  1851: 

At  Washington  (partial  eclipse) : 

Beginning  of  eclipse,  July  27,  19*  21-  3K2     M.T. 

End  «  "      "    20   50    38  .0        « 

At  Konigsbcrg  (total  eclipse)  : 

Beginning  of  eclipse,  July  28,  3  38  10 .8  " 

Beginning  of  total  obsc.,  "      "  4  38  57 .6  " 

End  of  total  obscuration,  "      "  4  41  54 .2  « 

End  of  eclipse,  "      "  5  38  32 .9  « 


LONGITUDE. 


535 


For  these  places  we  have  given — 


Washington, 
Konigsberg, 


Lat.  0 

+  38°  53'  39".25 
+  54    42  50  .4 


Long,  u 

+  5*    8-  11-.2 
—  1   22      0  .4 


The  longitudes  are  reckoned  from  Greenwich.  That  of 
Konigsberg  will  be  assumed  as  correct,  while  that  of  Washington 
will  be  regarded  as  an  approximate  value  which  it  is  proposed 
to  correct  by  these  observations. 

I.  The  mean  Greenwich  time  of  conjunction  of  the  sun  and 
moon  in  right  ascension  being,  July  28,  2A  21wl  2*. 6,  the  general 
eclipse  tables  will  be  constructed  for  the  Greenwich  hours  0A,  1*, 
2*,  3A,  4A,  and  5*  of  July  28.  For  these  times  we  find  the  follow- 
ing quantities  from  the  Nautical  Almanac  : 

For  the  Moon.* 


Greenwich  mean 
time. 

a 

6 

77 

July  28,  0* 

125°  40'    6".75 

+  20°    3'30".00 

60'  27".30 

1 

126    19    9  .41 

19    58    9  .36 

28  .41 

2 

126    58  10  .80 

19    52  39  .99 

29  .49 

3 

127   37  10  .82 

19    47     1  .92 

30  .54 

4 

128    16    9  .37 

19   41  15  .21 

31  .56 

5 

128-55    6  .36 

19    35  19  .89 

32  .56 

For  the  Sun. 


Greenwich  mean 
time. 

a' 

6' 

logr' 

July  28,  0A 

127°    6'    5".25 

-f  19°  5'  24".70 

0.006578 

1 

8  32  .63 

4  50  .23 

76 

2 

10  59  .99 

4  15  .74 

74 

3 

13  27  .34 

3  41  .21 

72 

4 

15  54  .67 

3     6  .64 

70 

5 

18  21  .99 

2  32  .05 

67 

*  The  moon's  a  and  6  in  the  Naut.  Aim.  are  directly  computed  only  for  every  noon 
and  midnight  and  interpolated  for  each  hour.  I  have  not  used  these  interpolated 
values,  but  have  interpolated  anew  to  fifth  differences.  The  moon's  parallax  has 
been  diminished  by  0".3  according  to  Mr.  ADAMS'S  Table  in  the  Appendix  to  the 
Naut.  Aim.  for  1856. 


536  SOLAR    ECLIPSES. 

With,  these  values  we  form  the  following  tables,  as  in  Art.  297 


a 

d 

Exterior  Contacts. 

Interior  Contacts. 

I 

y 

I 

logt 

c* 

127°    6'  17".22 

19°  5'  16".56 

0.534046 

7.663244 

—  0.011771 

7.661131 

1 

8  39  .51 

4  42  .76 

4023 

45 

11795 

32 

2 

11     1  .78 

4     8  .96 

3973 

47 

11844 

34 

3 

13   24  .03 

3  35  .14 

3899 

49 

11917 

36 

4 

15  46  .27 

3     1  .30 

3801 

51 

12015 

38 

5 

18     8  .50 

2  27  .46 

3679 

53 

12137 

40 

X 

AI 

*f 

A, 

y 

A! 

A, 

A3 

+  12 
+  9 
+  2 

0* 
1 
2 
3 
4 
5 

—  1.338900 
—  0.769-366 
—  0.199775 
+  0.369815 
+  0.939350 
+  1.508766 

+  0.569534 
.569591 
.569590 
.569535 
.569416 

+  67 

—  55 
—119 

—58 
—54 
—64 

!+  0.968589 
.885569 
.802185 
.718449 
.634370 
.549950 

—  0.083020 
.083384 
.083736 
.084079 
.084420 

-364 
—352 
—  343 
-341 

Hence  the  mean  changes  x'  and  yf,  for  the  epoch  T0  =  2*  (ac- 
cording to  the  method  of  Art.  296),  and  the  corresponding  values 
of  N  and  log  n,  are  as  follows  : 


z' 

y' 

N 

log  n 

0* 

-f  0.569563 

—  0.083202 

98°  18'  39".7 

9.760126 

1 

591 

3384 

19  42  .7 

168 

T  9 

Jo  —  -' 

600 

3562 

20  45  .3 

194 

3 

590 

3736 

21  47  .5 

205 

4 

563 

3908 

22  50  .0 

203 

5 

514 

4078 

23  52  .V 

186 

II.  The  full  computation  for  Konigsberg,  where  both  exterior 
and  interior  contacts  were  observed,  will  serve  to  illustrate  the 
use  of  the  preceding  formulae  in  every  practical  case. 

For  <p  =  54°  42'  50".4  we  find 


log?  sin  <?>'=  9.909898 


log  p  cos  /—  9.762639 


The  sidereal  time  at  Greenwich  mean  noon,  July  28,  was 
8*  22™  13*. 27,  with  which  //  is  found  as  given  below.  The  com- 
putation of  c,  37,  and  L  will  be  as  follows : 


LONGITUDE. 


537 


1st  Ext.  Cont. 

1st  Int.  Cont. 

2d  Int.  Cont. 

2d  Ext.  Cont. 

t 

3*  38™  10*.  8 

4*  38'*  57'.  6 

4*  41"»  54*.  2 

5*  38"*  32'.  9 

t  +  w 

2  16  10.4 

3  16  57.2 

3  19  53.8 

4  16  32.5 

ft 

12  0  46.44 

13   1  43.22 

13  4  40.31 

14  1  28.31 

ft  (in  arc) 

180°  11'  36".6 

195°  25'  48".  3 

196°  10'  4".  7 

210°  22'  4".  7 

For  t  4-  w,  a 

127  11  40  .1 

127  14  4  .2 

127  14  11  .2 

127  16  25  .6 

"       d 

19   3  59  .8 

19   3  25  .6 

19   3  23  .9 

19   2  52  .0 

li—  a 

52  59  56  .5 

68  11  44  .1 

68  55  53  .5 

83   5  39  .1 

log  sin  (fj.  —  a) 

9.902343 

9.967762 

9.969952 

9.996838 

log  cos  (ft  —  a) 

9.779473 

9.569889 

9.555679 

9.080040 

log  f 

9.664982 

9.730401 

9.732591 

9.759477 

f 

4-  0.462362 

4-  0.537528 

4-  0.540244 

4-  0.574748 

log  ^4  sin  B 

9.909898 

9.909898 

9.909898 

9.909898 

log  J.  cos  B 

9.542112 

9.332528 

9.318318 

8.842679 

B 

66°  47'  32".2 

75°  10'  40".  4 

75°  38'  5".9 

85°  6'  14".  3 

B  —  d 

47  43  32  .4 

66   7  14  .8 

56  34  42  .0 

66   3  22  .3 

log  ^ 

9.946544 

9.924595 

9.923693 

9.911486 

log  sin  (5  —  rf) 

9.869192 

9.919191 

9.921499 

9.960919 

log  cos  (B  —  d) 

9.827809 

9.746201 

9.740991 

9.608355 

log  T) 

9.815736 

9.843786 

9.845192 

9.872405 

n 

4-  0.654239 

4-  0.697888 

4-  0.700152 

4-  0.745427 

logC 

9.774353 

9.670796 

9.664684 

9.519841 

For  t  4-  w,    log  i 

7.663248 

7.661137 

7.661137 

7.663252 

I 

4-  0.533956 

—  0.011940 

—  0.011944 

4-  0.533772 

»s 

4-  0.002739 

4-  0.002148 

4-  0.002117 

+  0.001524 

Z 

4-  0.531217 

—  0.014088 

—  0,014061 

4-  0.532248 

III.  The  epoch  of  the  table  of  xf  and  yf  being  TQ=  2*,  we  have 
for  this  time 


xn=  —  0.199775 


=       0.802185 


with  which  we  proceed  to  find  the  values  of  co. 


m  sin  M  =  XQ  —  f 

—  0.662137 

—  0.737303 

—  0.740019 

—  0.774523 

m  cos  M  =  ya  —  r) 

4-  0.147946 

4-  0.104297 

4-  0.102033 

4-  0.056758 

log  m  sin  M 

n9.  820948 

n9.867646 

n9.  869242 

n9.  889035 

log  m  cos  M 

9.170107 

9.018272 

9.008741 

8.754027 

M 

282°  35'  42".  8 

278°  3'  5".4 

277°  51'  1".5 

274°  11'  28".  3 

log  m 

9.831527 

9.871949 

9.873331 

9.890198 

For  t  4-  o,     N 

98°  21'  2".l 

98°  22'  5".l 

98°  22'  8".2 

98°  23'  7".  3 

11   "      log  n 

9.760198 

9.760206 

9.760206 

9.760200 

M—  N 

184°  14'  40".  7 

179°  41'  0".3 

179°  28'  53".3 

175°  48'  21".0 

log  sin  (M—  N) 

n8.869321 

7.742363 

7.956643 

8.864135 

538 


SOLAR    ECLIPSES. 


log  L 

9.725272 

7*8.148849 

w8.148016 

9.726114 

log  sin  4, 

n8.975576 

n9.465463 

n9.681958 

9.028219 

4 

185°  25'  27".  7 

343°    1'    8".  6 

208°  44'  14".0 

6°    7'  33".  2 

M  —  N  —  4 

358    49  13  .0 

196    39  51  .7 

330    44  39  .3 

169    40  47  .8 

iogsin^/  —  JV  —  4/) 

718.313626 

7*9.457526 

n9.689051 

9.2532C8 

A  =  3600,         log  A 

3.556303 

log  r 

2.965682 

3.660109 

3.676521 

3.911290 

r 

4-  0A15W24*.0 

4-  P  16™  12'.  0 

4-  1*19"*   8M 

4-  2*15'»52'.5 

*;•*> 

-  1   38    10.8 

—  2  38    57  .6 

—  2  41    54.2 

—  3  38    32  .9 

0) 

—  1  22    46.8 

—  1   22    45.6 

—  1   22    46.1 

—  1  22    40  .4 

IV.  Equations   of  condition.  —  To   find  7\  and   *,  we   have  for 


whence 

/y>      QI  j-|     ~\T 

' 


log  XQ  =  n9.3006 
log  yQ  =    9.9043 


N=  98°  20'.7 
log  n  —  9.7602 


=  +  0.3434         -x0  cos  ^=  —  0.0290     log  H  =  2.9822 


.=  +  0.2023 
=     2*.5457 

=   3630" 


4-  y0  sin  N  =  -f  0.7938     log  r'     =  0.0066 
x  —  _p  0.7648     log  :r     =  3.5599 


log  x  =       9.8835     log  —  =  9.4157 
r'  Tt 


log  /5  =  log  ^     '-  =  9.9128 
1  —  ee 


log  v  =  log  —  =  0.2362 


With  these  constants  prepared,  we  readily  form  the  coefficients 
of  the  equations  of  condition  as  follows : 


log  tan  ^ 
log  sec  -ty 
v  tan  4, 
v  sec  4, 

1st  Ext.  Cont. 

1st  Int.  Cont. 

2d  Int.  Cont. 

2d  Ext.  Cont. 

8.9775 
nO.0019 
4-  0.163 
—  1.730 

7*9.4848 
0.0194 
—  0.526 
4-  1.801 

9.7390    ' 
nO.0571 
4-  0.944 
—  1.964 

9.0307 
0.0025 
4-  0.185 
4-  1.733 

v  (!?"-£> 

—  C*.2762 
7*9.4412 

4-  0*.7355 
9.8666 

4-  C\7860 
9.8954 

4-  P.7300 
0.2380 

vn(t  4-  w  —  7\) 
—  xv  tan  4, 

—  0.2739 
-  0.1251 

4-  0.7295 
4-  0.4023 

4-  0.7795 
—  8.7223 

4-  1.7155 
—  0.1414 

_  v  sec  ^ 
E 

4-  0.4506 

—  0.4691 

4-  0.5117 

-  0.4512 

4-  0.0516 

4-  0.6627 

4-  0.5689 

4-  1.1229 

LONGITUDE. 


539 


xv  tan 
L  v  sec 


log 


log  1st  part  of  F 

log  cos  (N  +  4,) 
log  ( —  vj3  cos  d  sec  4/) 

log  2d  part  of  F 
1st  part  of  F 
2d  »  «  ^ 


1st  Ext.  Cont. 

1st  Int.  Cont. 

2J  Int.  Cont. 

2d  Ext.  Cont 

—  0.2739 

+  0.7295 

+  0.7795 

-f  1.7155 

-  0.1251 

+  0.4023 

—  0.7223 

-  0.1414 

+  0.9192 

-f  0.0254 

—  0.0276 

-  0.9222 

+  0.5202 

-f  1.1572 

-f  0.0296 

+  0.6519 

9.7162 

0.0634 

8.4713 

9.8142 

9.5246 

- 

9.2408 

9.5880 

7.9959 

9.3388 

283°  46'.2 

81°  21'.8 

307°  4'.  9 

104°  28'.  3 

9.3766 

9.1766 

9.7803 

n9.3978 

0.1264 

nO.1439 

0.1816 

nO.1270 

9.5030 

n9.3205 

9.9619 

9.5248 

-f  0.1741 

-f  0.3873 

-f  0.0099 

-f  0.2182 

4-  0.3184 

-  0.2092 

+  0.9160 

+  0.3349 

-f  0.4925 

+  0.1781 

+  0.9259 

-j-  0.5531 

Putting  to'  -f  vf  =  J2,  we  have,  therefore,  for  the  four  Konigs- 
berg  observations,  the  equations 


(A) 


12=:  — 1*22">  46'.8  + 0.163 1?  — 1.730  TrAA;  — 1.730 -^+0.052  ATT  + 0.493  TTA^ 
i2  =  — 1    22   45.6—0.526     —1.801 


Q  =  — 1    22   46.1  +  0.944     +1.964 
12  =  —  1    22   40.4  +  0.185     +1.733 


+  1.801 
—  1.964 
+1.733 


+  0.663 
+  0.569 
4-1.123 


+  0.178 
+  0.926 
+  0.553 


where  we   have   annexed   a   column   for  the  weight  p,  giving 
interior  contacts  double  weight. 

A  similar  computation  for  the  two  observations  at  Washington 
gives  the  following  equations,  in  which  Q' =  o/'-f-  v^,  ID"  de- 
noting the  true  longitude  of  Washington : 


1.660  #  —  2.3927TA&  —  2.392  —  —  2.681  ATT  +  0.722 


Q'==  5  7    21.9  —  2.406     +2.959         +2.959         +0.509 


1.323 


More  observations  would  be  necessary  in  order  to  determine 
all  the  corrections ;  but  I  shall  retain  all  the  terms  in  order  to 
illustrate  the  general  method.  Subtracting  each  of  the  Konigs- 
berg  equations  from  each  of  those  which  follow  it,  we  obtain  the 
six  equations, 


0.315 


540  SOLAR    ECLIPSES. 

p 

i     0  =  4-  1*.2  —  0. 689  #—  0.071  TA/C  4-  8.681™  4-  0.611  AT  — 

l    0  =  4-0.74-  0.781  -f  3.694  -  0.234  +  0.517  4-  0.433 

0  =  4-6.44-0.022  4-3.463  4-3.463  4-1.071  4-0.060 

0  =  _  0  .5  4-  1.470  4-  3.765  -  3.765  -  O.C94  4-  0.748 

0  =  4-5.24-  0.711  4-  3.534  -  0.068  -f  0.460  4-  0.375 

0  ==  -f  5  .7  —  0.759  -  0.231  4-  3.697  4-  0.554  -  0.373 

where  the  weight  in  each  case  is  the  quotient  obtained  by 
dividing  the  product  of  the  two  weights  of  the  equations  whose 
difference  is  taken,  by  the  sum  of  the  weights  of  the  four 
original  equations  (Art.  334). 

The  same  method,  applied  in  the  case  of  the  two  Washington 
equations,  gives  the  single  equation 


0  =  _  8'.0  —  4.066  &  4-  5.351  TA&  4-  5.351    —  -f-  3.190  AT  —  2.055  7rA<?e 

r 

From  the  equations  (A')  and  (B ')  are  formed  the  following 
final  equations,  having  regard  to  their  weights,  in  the  usual 
manner : 

0  =  4-  15.495  4-  10.426tf  —    5.300  TA/fc  —  16.377  ^  —    6.609  AT  -f-  5.281 

0  =  —  12.445  —    5.300  4-  34.506  4-    6.135  4-  10.040  —  2.575 

0  =  _    8.191  _  16.377  4-    6.135  4-34.595  -f- 10.740  -8.214 

0  =  —    9.371  —    6.609  4-  10.040  4-  10.740  4-    5.672  —  3.316 

0=4-    7.951  4-    5.281  -    2.575  -    8.214  -    3.316  4-  2.675 

As  we  cannot  expect  a  satisfactory  determination  of  ATT  and 
from  these  observations,  we  disregard  the  last  two  equations, 

and  then,  solving  the  first  three,  we  obtain  $,  TTA/.:,  and —j- in 
terms  of  ATT  and  TTACC,  as  follows  : 


=  —  4".36  4-  0.375  ATT  —  0.525  x 
=  -f  0  .02  —  0.216  A*  —  0.004 

— =  —  1  .83  —  0.095  ATT  —  0.010 


These  values  substituted  in  the  equations  (A)  give 

Q  =  _  1*  22-  44'.38  -f  0.651  A*  -f  0.432 
fl  =  —  1  22  46  .64  4-  0.684  4-  0.443 
fl  =  —  1  22  46  .58  4-  0.685  4-  0.442 
fl  ==  —  1  22  44  .34  +  0.653  4-  0.432 


LONGITUDE.  541 

the  mean  of  which,  giving  the  second  and  third  double  weight,  is 
(A")  Q  =  —  l»  22™  45'.86  -j-  0.674  ATT  +  0.439 

The  equations  (B)  become 

Q'=  5*  7"  26«.99  —  1.314  Ar  —  0.116  r 
&=  5   7    27  .03  —  1.314         -  0.101 

the  mean  of  which  is 

(B")  &=  5*  7™  27«.01  —  1.314  Ar  —  0.109  r 


Now,  if  we  assume  the  longitude  of  Konigsberg  to  be  well 
determined,  we  have 

fl  =  at'  -f  vr  =  —  1»  22m  0*.4  -f  vf 

which,  with  the  equation  (A"),  gives 

vr  =  —  45'.46  -f-  0.674  ATT  -f  0.439  n&ee 
Hence,  by  (B  "),  the  true  longitude  of  Washington  is 
w"=  Q'  —  v    ==  5*  8-  12'.47  —  1.988  A*  —  0.548 


If  the  longitude  of  Washington  were  also  previously  well  estaK 
lished,  this  last  equation  would  give  us  a  condition  for  deter- 
mining the  correction  of  the  moon's  parallax.  Thus,  if  we  adopt 
a)"=5h  8m  12*.  34,  which  results  from  the  U.S.  Coast  Survey 
Chronometric  Expeditions  of  1849,  '50,  '51,  and  '55,  this  equation 
gives 

0^-f-  0.13  —  1.988  Ar  —  0.548 
whence 

A*  =       0".07  —  0.276 


The  probable  value  of  Aee,  according  to  BESSEL,  is  within 
±  0.0001,  so  that  the  last  term  cannot  here  exceed  0".10.  If, 
therefore,  the  above  observations  are  reliable  and  the  supposed 
longitudes  exact,  the  probable  correction  of  the  parallax  indi- 
cated scarcely  exceeds  O'M,  a  quantity  too  small  to  be  established 
by  so  small  a  number  of  observations.  Nevertheless,  the  example 
proves  both  that  the  adopted  parallax  is  very  nearly  perfect,  and 
that  a  large  number  of  observations  at  various  well  determined 
places  in  the  two  hemispheres  may  furnish  a  good  determination 
of  the  correction  which  it  yet  requires. 


542 


LUNAR    ECLIPSES. 


Finally,  the  corrections  of  the  Ephemeris  in  right  ascension 
and  declination,  according  to  the  above  determination  of  7-  and 
$,  are  found  by  (586)  to  be  (putting  a'  for  a  and  of  for  d) 

A(O  —  »')  =  —  28".42  +  0.469  Ar  -f-  0.187 
A(<5  _  <j')  =  -  •    0  .48  +  0.314  ATT  —  0.556 

This  large  correction  in  right  ascension  agrees  with  the  results 
of  the  best  meridian  observations  on  and  near  the  date  of  this 
eclipse.  Since  that  time  the  Ephemerides  have  been  greatly 
improved. 

LUNAR    ECLIPSES. 

.  To  find  whether  near  a  given  opposition  of  the  moon  and  sun 
a  lunar  eclipse  will  occur.  —  The  solution  of  this  prob- 
lem is  similar  to  that  of  Art.  287,  except  that  for 
the  sun's  semidiameter  there  must  be  substituted  the 
apparent  semidiameter  of  the  earth's  shadow  at  the 
distance  of  the  moon  ;  and  also  that  the  apparent 
distance  of  the  centres  of  the  moon  and  the  shadow 
will  not  be  affected  by  parallax,  since  when  the 
earth's  shadow  falls  upon  the  moon  an  eclipse  occurs 
for  all  observers  who  have  the  moon  above  their 
horizon.  If  $,  Fig.  45,  is  the  sun's  centre,  E  that 
of  the  earth,  LM  the  semidiameter  of  the  earth's 
shadow  at  the  moon,  we  have 

Apparent  semidiameter  of  the  total 

shadow  =  LEM 

*=  BLE—EVL 

=  BLE—(AES  —  EAV) 


<vhere  we  employ  the  same  notation  as  in  Art.  287. 

But  observation  has  shown  that  the  earth's  atmosphere 
increases  the  apparent  breadth  of  the  shadow  by  about  its  one- 
fiftieth  part:*  so  that  we  take 

*  This  fractional  increase  of  the  breadth  of  the  shadow  was  given  by  LAMBERT  as 
^,  and  by  MAYER  as  -£$.  BEER  and  MADLER  found  ^  from  a  number  of  observations 
of  eclipses  of  lunar  spots  in  the  very  favorable  eclipse  of  December  26,  1833.  See 
"  Der  Mond  nach  seinen  kosmischen  und  individuellen  Verhaltnissen,  oder  allgemeint 
vergleichcndc  Selenographie,  von  WILHELM  BEER  und  DR.  JOHANN  HEINRICH  MADLER," 
§98. 


LUNAR   ECLIPSES.  543 

App.  semid.  of  shadow  =  —  (r  —  s'  -j-  r')  (587) 

In  order  that  a  lunar  eclipse  may  happen,  we   must  have, 
therefore,  instead  of  (477), 

/3  cos  /'<  —  (TT  —  s'+  ;:')  +  5  (588) 

50 

or,  taking  a  mean  value  of  1',  as  in  Art.  287, 

(7r  ~  s'  +  ^  "H  x  L00472 

Employing  mean  values  in  the  small  fractional  part,  we  have 

-f  si  x  .00472  =  16" 
J 

and  the  condition  becomes 

+  «  +  w"  (589) 


If  in  this  we  substitute  the  greatest  values  of  TT,  nf,  and  s,  and 
the  least  value  of  s',  the  limit 

0  <  63'  53" 

is  the  greatest  limit  of  the  moon's  latitude  at  the  time  of  opposi- 
tion for  which  an  eclipse  can  occur. 

If  we  substitute  the  least  values  of  IT,  TT',  and  5,  and  the  greatest 
value  of  s',  the  limit 

/3<52'4" 

is  the  least  limit  for  which  an  eclipse  can  fail  to  occur. 

Hence,  a  lunar  eclipse  is  certain  if  at  full  moon  ft  <  52'  4", 
impossible  if  ft  >  63'  53",  and  doubtful  between  these  limits.  The 
doubtful  cases  can  be  examined  by  (589),  or  still  more  exactly 
by  (588),  employing  the  actual  values  of  TT,  TT',  5,  5',  at  the  time, 
and  computing  I'  by  (475). 

These  limits  are  for  the  total  shadow.  For  the  penumbra  we 
have 

App.  semid.  of  penumbra  =  —  (r  -f  s'  -j-  TT')  (590) 


544  LUNAR   ECLIPSES. 

so  that  the  condition  (588)  may  be  employed  to  determine 
whether  any  portion  of  the  penumbra  will  pass  over  the  moon, 
by  substituting  -f  s'  for  —  s'.  It  will  be  worth  while  to  make 
this  examination  only  when  it  has  been  found  that  the  total 
shadow  does  not  fall  upon  the  moon. 

339.  To  find  the  tim.e  when  a  given  phase  of  a  lunar  eclipse  will 
occur. — The  solution  of  this  problem  may  be 
derived  from  the  general  formulae  given  for 
solar  eclipses,  by  interchanging  the  moon  and 
earth  and  regarding  the  lunar  eclipse  as  an 
eclipse  of  the  sun  seen  from  the  moon  ;  but  the 
following  direct  investigation  is  even  more 
simple. 

Let  $,,Fig.  46,  be  the  point  of  the  celestial 
sphere  which  is  opposite  the  sun,  or  the  appar- 
ent geocentric  position  of  the  centre  of  the 

earth's  shadow;   M,  the  geocentric  place  of  the  centre  of  the 

moon  ;  P,  the  north  pole.     If  we  put 

a  =  the  right  ascension  of  the  moon, 
a'  —  the  right  ascension  of  the  point  S, 

=  R.  A.  of  the  sun  -f  180°, 
3   =  the  declination  of  the  moon, 
d'  =  the  declination  of  the  sun, 
Q  =  the  angle  PSM, 
L  =  8M, 
we  have 

—  d'=  the  declination  of  S, 


and  the  triangle  PSM  gives 

sin  L  sin  Q  —  cos  d  sin  (a  —  a') 

sin  L  cos  Q  =  cos  3'  sin  d  -f-  sin  3'  cos  8  cos  (a 


^ 
-O     } 


The  eclipse  begins  or  ends  when  the  arc  SMis  exactly  equal  to 
the  sum  of  the  apparent  semidiameters  of  the  moon  and  the 
shadow.  The  figure  of  the  shadow  will  differ  a  little  from  a 
circle,  as  the  earth  is  a  spheroid ;  but  it  will  be  sufficiently  accu- 
rate to  regard  the  earth  as  a  sphere  with  a  mean  radius,  or  that 
for  the  latitude  45°.  This  is  equivalent  to  substituting  for  TT  in 
(587)  and  (590)  the  parallax  reduced  to  the  latitude  45°,  which 
may  be  found  by  the  formula 


LUNAR   ECLIPSES.  545 

7^  =  [9.99929]  TT  (592) 

where  the  factor  in  brackets  is  given  by  its  logarithm. 

Hence  the  first  and  last  contacts  of  the  moon  with  the  pe- 
numbra occur  when  we  have 

i  =  S  (-,  +  «'+«')  +  *  (593) 

ou 

For  the  first  and  last  contacts  with  the  total  shadow, 

L  =  ~(.^  -«'+«')  +  «  (594) 

Ov 

For  the  first  and  second  internal  contacts  with  the  penumbra, 

')-«  (595) 


For  the  first  and  second  internal  contacts  with  the  total  shadow, 
or  the  beginning  and  end  of  total  eclipse, 

£  =  |i  (*,-«'+*')-*  (596) 

The  solution  of  our  problem  consists  in  finding  the  time  at 
which  the  equations  (591)  are  satisfied  when  the  proper  value  of 
L  is  substituted  in  them.  A  very  precise  computation  would, 
however,  -be  superfluous,  as  the  contacts  cannot  be  observed  with 
accuracy,  on  account  of  the  indefinite  character  of  the  outline 
both  of  the  penumbra  and  of  the  total  shadow.  It  will  be  suffi- 
cient to  write  for  (591)  the  following  approximate  formula,  easily 
deduced  from  them  : 


\ 
\ 


L  sin  Q  =  (a  —  a')  cos  d 

Let  us  put 

_  sin  2  8  sin'  \  (a  —  a') 

sin  1" 

(598) 


_  sin  2  9  sin'  fr  (a  —  a') 
sin  V 

X  =  (a  —  a')  COS  d 
y=d  -f  3'—  e 
xf,  y'=  the  hourly  increase  of  x  and  y  , 


then,  if  the  values  of  x  and  y  are  computed  for  several  successive 

VOL.  L—  35 


546  LUNAR   ECLIPSES. 

hours  near  the  time  of  full  moon,  we  shall  also  have  xr  and  y' 
from  their  differences;  and  if  x0  and  y0  denote  the  values  of  x 
and  y  for  an  assumed  epoch  7J,,  near  the  time  of  opposition,  we 
shall  have  for  the  required  time  of  contact  T—  TQ+  T  the 
equations 

L  sin  Q  =  XQ  -{-  x'r 

from  which  r  is  obtained  by  the  process  already  frequently 
employed  in  the  preceding  problems.  Thus,  computing  the 
auxiliaries  m9  M9  n,  N,  by  the  equations 

m  sin  M  =  xn  nsinJV=#'  )     /con\ 

,..  __        .  f    (oyyj 

m  cos  M  ==  y0  n  cos  J\r  =  y  ) 

we  shall  have 

m  sin  (M  —  N) 


8in  4  = 


L 

_  L  cos  4       m  cos  (M  —  N) 

T"  ~~  '  -1-"- --'-—-• 


(600) 


in  which  we  take  cos  ^  with  the  negative  sign  for  the  first  contact 
and  with  the  positive  sign  for  the  last  contact. 

The  angle  Q==N-\-^  is  very  nearly  the  supplement  of  the 
angle  PMS,  Fig.  46  ;  from  which  we  infer  that  the  angle  of  posi- 
tion of  the  point  of  contact  reckoned  on  the  moon's  limb  from  the  north 
point  of  the  limb  towards  the  east  —  180°  +  N  -{-  $. 

The  time  of  greatest  obscuration  is  found,  as  in  Art.  324,  to  be 


(601) 


which  is  also  the  middle  of  the  eclipse. 

The  least  distance  of  the  centres  of  the  shadow  and  of  the 
moon  being  denoted  by  J,  we  have,  as  in  Art.  324, 

A  =  ±  m  sin  (  M  —  N)  (602) 

the  sign  being  taken  so  that  J  shall  be  positive.     If  then  we  put 

D  =  the  magnitude  of  the  eclipse,  the  moon's  diameter  being 
unity, 


LUNAR    ECLIPSES. 


54; 


we  evidently  have 


D  = 


L  —  J 
2s 


(603) 


in  which  the  value  of  L  for  total  shadow  from  (594)  is  to  be 
employed. 

The  small  correction  e  in  (598)  may  usually  be  omitted,  but 
its  value  may  be  taken  at  once  from  the  following  table : 

Value  of  f. 


,5 

a  —  a' 

0" 

1000" 

2000" 

3000" 

4000" 

5000" 

6000" 

0° 

0" 

0" 

0" 

0" 

0" 

0" 

0" 

5 

0 

0 

1 

2 

3 

5 

8 

10 

0 

0 

2 

4 

7 

10 

15 

15 

0 

1 

2 

6 

10 

15 

22 

20 

0 

1 

3 

7 

13 

19 

28 

25 

0 

i 

4 

8 

15 

23 

33 

30 

0 

1 

4 

9 

17 

26 

38 

The  quantity  e  has  the  same  sign  as  <5,  and  is  to  be  subtracted 
algebraically  from  d  -f-  d'. 

EXAMPLE. — Compute  the  lunar  eclipse  of  April  19,  1856.  The 
Greenwich  mean  time  of  full  moon  is  April  19,  21*  5W1.5.  We 
therefore  compute  the  co-ordinates  x  and  y  for  the  Greenwich 
times  April  19,  18",  21*,  24*. 


18* 

21* 

24* 

>R.A.                     -=o 

13*  46"  36'.62 

13*52-    9-.81 

13*57W45M2 

OR  A.  +  180°  =  a' 

13  52    52.98 

13  53    20.93 

13  53    48.88 

a—  a' 

—     6    16.36 

—     1    11.12 

+     3    56.24 

a  —  a'  (in  arc) 

-  5645" 

-  1067" 

+     3544" 

D  Decl.  =  (5 

—11°  27'   0".2 

-12°    6'43".7 

-12°  46'    5".5 

Q     «     =  (*' 

+11    35  49  .4 

+11    38'  22  .8 

+11  40  56  .6 

—  e 

+             13. 

0  . 

+               6. 

!/ 

+           542" 

1701" 

3903" 

log  (a  —  a') 

n3.75166 

n3.02816 

3.54949 

log  COS  d 

9.99127 

9.99022 

9.98913 

log* 

W3.74293 

n3.01838 

3.53862 

548  LUNAR   ECLIPSES. 

Hence  we  have  the  following  table : 


Diff. 

Diff. 

X 

y 

18* 
21 
24 

—  5533" 
—  1043 
4-3456 

4-4490 
4-4499 

^=4-1498 

4-  542" 
—  1701 
—  3903 

—  2243 
—  2202 

,  —  ,« 

To  find  L,  we  have,  by  (593)  and  (594), 

TT  ^r  54'  32"      ^  =  3267" 
s' =    957 

n'  =        9 


^  _  S'  4-  TT'  =  2319 

(^  _  S'  4_  „')  =      46 

g=    891 

.£  for  shadow  =  3256 


^  4.  «'  4-  *'  =  4233" 

s  =    891    , 
i  for  penumbra  =  5209 


Assuming  the  time  T9  —  21A,  we  proceed  by  (599)  and  (600) : 


.r0  -.=  m  sin  M 

—  1043 

x'  =  n  sin  JV 

4-  1498 

y0  =  m  cos  M 

-1701 

y'  =  n  cos  JV 

741 

M 

210°  Sl'.O 

JV 

116°  19'.2 

log  m 

3.3000 

logw 

3.2230 

-cos(Jf—  JVT)^ 


=      21 


T.  =  Time  of  middle  of  eclipse  ==      21  .108 


log  sin  4 

L  cos  4 

n 

r, 

Beginning 
End 


Shadow. 

Penumbra. 

9.7855 

9.5815 

q=     1».542 

ip    2».881. 

21  .108 

21  .108 

19  .566 

18  .227 

22  .650 

23  .989 

For  the  magnitude  of  the  eclipse,  we  have,  by  (602)  and  (603) 


OCCULTATIONS    OF   FIXED    STARS.  549 

m  sin  (M  —  N)  =  J  =  1987" 


1st  Contact. 

2d  Contact. 

* 

142°  24' 

37°  36' 

N 

116    19 

116    19 

180°+  N+  * 

78    43 

333    55 

£  -  J  =  1269  ~  1782 

2s  =  1782 

For  the  position  of  the  points  of  contact  with  the  shadow,  we 
have,  from  the  above  value  of  log  sin  $  for  shadow,  taking  cos  ^ 
as  negative  for  first  and  positive  for  second  contact, 


and  hence 

1st  contact  is  79°  from  north  point  of  limb  towards  the  east, 
2d  26°     "        «          «  «  «  west. 

The  times  of  the  several  contacts  for  any  meridian  are  obtained 
from  the  times  above  found  by  subtracting  the  west  longitude  of 
that  meridian. 

OCCULTATIONS    OF   FIXED    STARS. 

340.  The  occultation  of  a  fixed  star  by  the  moon  may  be 
treated  as  a  simple  case  of  a  solar  eclipse,  in  which  the  sun  is 
removed  to  so  great  a  distance  that  its  parallax  and  semidiameter 
may  be  put  equal  to  zero.  The  cone  of  shadow  then  becomes 
a  cylinder,  and  the  point  Z  of  Art.  289  is  nothing  more  than 
the  position  of  the  star,  so  that  the  co-ordinates  of  the  moon  at 
any  time  are  found  by  the  formulae  (482)  by  regarding  a  and  d 
as  the  right  ascension  and  declination  of  the  star.  In  like 
manner  the  co-ordinates  of  the  place  of  observation  will  be  found 
by  (483).  The  radius  of  the  shadow  is  constant  and  equal  to  A-, 
which  is,  therefore,  to  be  substituted  for  L  =  I  —  i£  in  (490)  and 
(491).  The  co-ordinates  z  and  £  will  not  be  required  unless  we 
compute  the  latter  for  the  purpose  of  taking  into  account  the 
effect  of  refraction  according  to  Art.  327. 

For  the  convenience  of  the  computer  I  shall  here  recapitulate 
the  formulae  required  in  the  practical  applications,  making  the 
modifications  just  indicated. 


550  OCCULTATIONS  OF  FIXED  STARS. 

841.  To  find  the  longitude  from  an  observed  occultation  of  a  star  by 
the  woon.^-According  to  the  method  of  Art.  329,  we  proceed  as 
follows: 

L  Find,  approximately,  the  time  of  conjunction  of  the  moon 
and  star  in  right  ascension,  reckoned  at  the  first  meridian.  Take 
from  the  Ephemeris,  for  four  consecutive  integral  hours,  two 
preceding  and  two  following  the  time  of  conjunction,  the  moon's 
right  ascension  (a),  declination  (o),  and  horizontal  parallax  ( x). 
Take  also  from  the  most  reliable  source  the  star's  right  ascension 
(a')  and  declination  (d'). 

For  each  of  these  hours  compute  the  co-ordinates  x  and  y  by 
the  formulae 

COS  <Jsin  (a  —  a') 

/v>  V '_ 

sin  TT 

__  sin  (3  —  d')  cos2  Ka  —  a')  4-  sin  (d  -f  dr)  sin2  j-  (a  —  a') 

sin  TT 

and,  arranging  their  values  in  a  table,  deduce  their  hourly 
variations  x'  and  y'  for  the  same  instant^. for  which  x  and  y  have 
been  computed. 

II.  Let  (i  be  the  local  sidereal  time  of  an  observed  immersion 
or  emersion  of  the  star  at  a  place  whose  latitude  is  <p,  and  west 
longitude  to ;  t  the  corresponding  local  mean  time.     The  co-or- 
dinates of  the  place  are  to  be  computed  by  the  formulae 

A  sin  B  =  p  sin  <p'  £  =  p  cos  <f>'  sin  (/*  —  a') 

A  cos  B  .=  p  cos  <p'  cos  (/i  —  a')  T]  =  A  sin  (B  —  J') 

Z  =  -A  cos  (5  —  <5') 

When  log  £  is  small,  add  to  logs  £  and  y  the  correction  for 
refraction  from  the  table  on  p.  517. 

III.  Assume  any  convenient  time   7^  reckoned   at  the   first 
meridian,  so  near  to  t  -f  to  that  x  and  y  may  be  considered  to 
vary  proportionally  with   the  time  in  the  interval  t  +  to  —  T0. 
For  the  assumed  time,  take  the  values  of  x  and  y  (denoting  them 
by  :r0  and  j/0),  and  also  those  of  x'  and  y1 ',  and  compute  the  aux- 
iliaries m,  M,  &c.  by  the  formulae 


LONGITUDE.  551 

m  sin  M  =  x0  —  £  n  sin  N  =  x' 

m  cos  M  =  y0  —  T?  n  cos  N  =  y' 

8in  +  =  ««i"(3f--y)  log  ,,  _  9.43500o* 

/t 

where  i}/  is  (in  general)  to  be  so  taken  that  cos  ij/  shall  be  nega- 
tive for  immersion  and  positive  *£or  emersion  (but  in  certain 
exceptional  cases  of  rare  occurrence,  and  of  but  little  use  in 
finding  the  longitude,  see  Art.  330).  Then 


_  hk  cos  ^       hm  cos  (M  —  N) 
n  n 


or,  when  sin  ^  is  not  very  small, 

hm   sin  (M  —  N  —  4) 


T.  =^= 


sin  4 


If  the  local  mean  time  t  was  observed,  take  h  =  3600  in  these 
formulae,  and  then  the  longitude  will  be  found  by 

But  if  the  local  sidereal  time  jj.  was  observed,  take  A  =  3609.856 
in  the  preceding  formulae;,  then,  //0  being  the  sidereal  time  at  the 
first  meridian  corresponding  to  7J, 

The  longitude  thus  found  will  be  affected  by  the  errors  of  the 
Ephemeris. 

IV.  To  form  the  equations  of  condition  for  correcting  the 
longitude  for  errors  of  the  Ephemeris  when  the  occultation  has 
been  observed  at  more  than  one  place,  compute  the  auxiliaries 

n 
x  =  —  #0  cos  N  -f-  ?/0  sin  N 

nit 
the  same  value  of  h  being  used  as  before. 

*  According  to  OUDEMANS  (Astron.  Nach.,  Vol.  LI.,  p.  30),  we  should  use  for  occul- 
tations  k  =  0.27264,  or  log  k  =  9.435590,  which  amounts  to  taking  the  moon's 
apparent  semidiameter  about  1".25  greater  in  occultations  than  in  solar  eclipses. 
But  it  is  only  for  the  reduction  of  isolated  observations  that  we  need  an  exact  value, 
since,  when  we  have  a  number  of  observations,  the  correction  of  whatever  value  of 
k  we  may  use  will  be  obtained  by  the  solution  of  our  equations  of  condition. 


552  ,  OCCULTATIONS    OF   FIXED    STARS. 

Then,  for  each  observation  at  each  place,  compute  the  coeffi- 
cients v  tau  ^  v  sec  i|/,  and 

E  =  vn  (t  +  w  —  TJ  —  x  v  tan  4- 

where  to  is  the  approximate  longitude  and  the  unit  of  t  -f  w  —  J'j 
is  one  mean  hour,  and  also 


in  which 

/3  =  £^  Jog  (1  —  «*)  =  9.99709 

Then,  to'  denoting  the  true  longitude, 

o/=  to  —  v?  -p  v  tan  4  .  #  -j-  v  sec  4,  .  TTA#  +  -^  •  A:r  + 
in  which  7-  and  t?  have  the  signification 

Y  =       sin  N  cos  <5  A(a  —  a')  -f-  cos  N  A(<S  —  <5') 
#  =  —  cos  N  cos  (J  A(a  —  a')  -j-  sin  N  A(«5  —  df) 

The  discussion  of  the  equations  of  condition  thus  formed  may 
then  be  carried  out  precisely  as  in  Art.  334,  taking  f,  $,  TTA/^,  A/T, 
and  x&ee  as  the  unknown  quantities. 

EXAMPLE.  —  The  occultation  of  Aldebaran,  April  15,  1850,  was 
observed  at  Cambridge,  Mass.,  and  Konigsberg,  as  follows:* 

At  Cambridge,   <p  =  42°  22'  48".6,  w  ==  4*  44-  30s. 

Immersion,   2*  lm  52'.45  Mean  time. 
Emersion,      3    1    38  .35       "         « 

-4*  Konigsberg,  <?  =  54°  42'  50".4,  w  =  -^  1»  22-  0-.4 

Immersion,    10*  57W  43«.66  Sidereal  time. 
Emersion,      11   47    47.60          «          " 

I.  The  Greenwich  mean  time  of  conjunction  of  the  moon  and 
star  was  about  7^  30W,  and  hence  we  take  our  data  from  the 
Nautical  Almanac  as  follows  : 

*  Astronomical  Journal,  Vol.  I.,  pp.  139  and  175. 


LONGITUDE. 


553 


1850  April  15. 

a 

j 

re 

6* 

65°  56'  21".16 

-|-  16°  40'    0".05 

58'  55".22 

7 

66    32  32  .06 

16    46  30  .53 

58  55  .87 

8 

67'     8  46  .02 

16    52  54  .77 

58  56  .49 

9 

67    45     3  .02 

16    59  12  .76 

58  57  .10 

The  position  of  Aldebaran  for  the  same  date  was 

*'=  66°  49'  33".9  d'=  +  16°  12'  1".7 

Hence,  by  I.  of  the  preceding  article,  we  form  the  following 
table : 


Gr.T. 

X 

*' 

y 

y' 

6* 

-0.86519 

+  0.58849 

4-  0.47664 

-f  0.10871 

7 

—  0.27671 

47 

.58531 

63 

8 

+  0.31176 

42 

.69390 

56 

9 

+  0.90014 

32 

.80243 

48 

II.  The  sidereal  time  of  Greenwich  Mean  Noon,  April  15, 
1860,  was  1*  33'"  8*. 96.  With  this  number,  converting  the 
Konigsberg  times  into  mean  times  for  the  sake  of  uniformity,  we 
find 


Cambridge. 

Konigsberg. 

Immersion. 

Emersion. 

Immersion. 

Emersion. 

2*    I™  52'.  45 

3*    1«»  38'.  35 

9*  23"  15'.64 

10*  13«  11'.88 

6    46    22.45 

7   46      8  .35 

8     1    15.24 

8   51     10.98 

54°    2'    2".  55 

69°    0'  58".35 

164°  25'  54".  90 

176°  56'  64".  00 

347    12    28  .65 

2    11    24  .45 

97    36   21  .00 

110      7  20  .10 

9.826441 

9.909898 

9.869121 

9.762639 

n9.214324 

8.451362 

9.758801 

9.735287 

9.646065 

9.641159 

9.904088 

9.922175 

9.944427 

9.952794 

9.185091 

8.549725 

ft  — a,' 

log  p  sin  f 

log  p  cos  0' 

log  I 

log  rj 


The  value  of  log  £  has  been  found  in  order  to  find  the  correc- 
tion for  refraction.  This  correction  is  here  quite  sensible  in  the 
case  of  the  Konigsberg  observations  which  were  made  at  a  great 


554 


OCCULTATIONS    OF    FIXED    STARS, 


zenith  distance.  By  the  table  on  p.  517,  we  find  that  the  loga- 
rithms of  f  and  y  must  be  increased  by  0.000006  for  immersion, 
and  by  0.000041  for  emersion.  Applying  these  corrections,  the 
values  of  the  co-ordinates  are  as  follows : 


—  0.16380 
4-  0.44266 


+  0.02827 
-f  0.43768 


4-  0.57386 
+  0.80175 


4-  0.54366 
4-  0.83602 


III.  Assuming  convenient  times  not  far  from  t  -{-  co,  we  have 


Assumed  T0 

6*.  8 

7*.  8 

8A.0 

8*.  85 

*o 

—  0.39440 

4-  0.19406 

4-  0.31176 

4-  0.81188 

y0 

4  0.56358 

4-  0.67218 

4  0.69390 

4  0.78615 

XQ  —  £  =  m  sin  M 

-  0.23460 

4-  0.16579 

—  0.26210 

+  0.26822 

ya  —  t]  =  m  cos  M 

4  0.12092 

4  0.23450 

—  0.10785 

—  0.04987 

M 

297°  40'  16".5 

35°  15'  36".  1 

247°  38'    1".0 

100°  31'  57".7 

log  m 

9.415608 

9.458164 

9.452433 

9.435871 

x'=  n  sin  N 

4  0.58847 

4  0.58843 

4  0.58842s 

4  0.58836 

y'  =  n  cosN 

4  0.10865 

4  0.10857 

4  0.10856 

4  0.10849 

N 

79°  32'  21".l 

79°  32'  45".8 

79°  32'  48".5 

79°  33'    8".  5 

4 

216    11  35  .9 

312    33  59  .0 

167    35  28  .5 

21      1  28  .1 

(h  =  3600)            r 

-  89'.  74 

-  128'.  82 

—  68'.  63 

—  3'.52 

u 

4*  44"*  37'.  81 

4A  44m  12i.83 

—  1*  22m    7'.  01 

—1*22"    4».90 

IV.  For  the  equations  of  condition,  taking  T0=  7*.  8, 


and  putting 


±  =       7fc.2772 
*  =  4-  0  .6258 


TT  =  3536" 
log  v  =  0.2308 


=  the  true  longitude  of  Cambridge, 
=  "  "  Konigsberg, 


we  find,  neglecting  terms  in  &ee, 

w^  ==  4*  44CT  37S.81  —  vr  4-  1.245  *  —  2.108  TTA£  —  1.293  Ar 
MI  ==  4  44  12  .83  —  vr  —  1.852  #  4-  2.515  TTA^  4-  1.660  ATT 
«i/=  —  1  22  7  .01  —  vy  —  0.374  *  —  1.742  TTA/<  4-  0.991  ATT 
«,/=  —  1  22  14  .90  —  vr  4-  0.654  #  4-  1.822  r^k  4-  1.195  A* 

whence  the  two  equations 

0  =  4-  24V98  4-  3.097  $  —  4.623  TTA*  —  2.953  ATT 
0  —  4-    7  .89  —  1.028  #  —  3.564  TTA^  —  0.204  ATT 

If  we  determine  $  and  TTA&  in  terms  of  A/T,  these  equations  give 
#  =  _  3".33  4-  0.607  ATT 

=  4-  3   .17  —  0.232  ATT 


LONGITUDE.  555 

and  then  we  find 

wj  =       4*  44-  26'.98  —  vr  —  0.048  A* 
<=  —  1   22    11.29  —  vr  +  1.169  ATT 

Assuming  &>/  =  —  1*  22™  0'.4  as  well  determined,  the  last  equa* 

tion  gives 

vr  =  —  KK89  +  1.169  ATT 

which  substituted  in  the  value  of  o^  gives 

wl  =  4*  44-  37'.87  —  1.217  ATT 

Finally,  adopting  the  correction  of  the  parallax  for  this  date  as 
given  in  Mr.  ADAMS'S  table  (Appendix  to  the  Nautical  Almanac 
for  1856),  ATI  =  -j-  5".l,  this  last  value  becomes 

wl  =  4*  44-  31-.66 

which  agrees  almost  perfectly  with  the  longitude  of  Cambridge 
found  by  the  chronometric  expeditions,  which  is  4h  44™  31*.95. 
With  the  same  value  of  ATT  we  find 


0  =  —  0".23  TTA£  =  +  1".99 

and  hence,  by  (586),  the  corrections  of  the  Ephemeris  on  this 
date,  according  to  these  observations,  are 

A(a  —a')  =  —  2".93  A(<5  —  3')  =  —  0".77 

The  value  TTA/<;  =  +  1".99  gives  A/J  =•  0.00056,  and  hence  the 
corrected  value  k  =  0.27227  -f  0.00056  =  0.27283,  which  is  not 
very  different  from  OUDEMANS'S  value.  (See  p.  551). 

v/ 

342.  "When  a  number  of  occultations  have  been  observed  at  a 

place  for  the  determination  of  its  longitude,  it  will  usually  be 
found  that  but  few  of  the  same  occultations  have  been  observed 
at  other  places.  If,  then,  we  were  to  depend  altogether  upon 
corresponding  observations  at  other  places,  we  should  lose  the 
greater  part  of  our  own.  In  order  to  employ  all  our  data,  we 
may  in  such  case  find  for  each  date  the  corrections  of  the  moon's 
place  from  meridian  observations  (see  Art.  235),  and,  employing 
the  corrected  right  ascension  and  declination  in  the  computation 
of  x  and  y,  our  equations  of  condition  will  involve  only  terms  in 
ir&k  and  A/T.  The  value  of  A;T  will,  however,  be  different  on 


556  OCCULTATIONS    OF    FIXED    STARS. 

different  dates,  and,  therefore,  if  we  wish  to  retain  this  term,  we 
must  introduce  in  its  stead  the  correction  of  the  mean  parallax 
which  is  the  constant  of  parallax  in  the  lunar  tables.  If  this 
constant  is  denoted  by  TTO,  we  shall  have,  very  nearly, 


A?r  =  —  Ar. 
• 

where  n  is  the  parallax  for  the  given  date.     The  equations  of 
condition  will  then  be  of  the  form 

wl  =  a>  -j-  a  .  Kbit  -\-  b  .  ATTO 

where 

a  =  v  sec  b  =.  —  E 


In  PEIRCE'S  Lunar  Tables,  now  employed  in  the  construction  of 
our  Ephemeris,  XQ=  3422".06, 

343.  The  passage  of  the   moon  through  a  well  determined 
group  of  stars,  such  as  the  Pleiades,  affords  a  peculiarly  favorable 
opportunity  for  determining  the  correction  of  the  moon's  semi- 
diameter  as  well  as  of  the  moon's  relative  place,  of  the  relative 
positions  of  the  stars  themselves,  and  also  (if  observations  are 
made   at   distant  but  well  determined  places)  of  the  parallax. 
Prof.  PEIRCE  has  arranged  the  formulae  of  computation,  with  a 
view  to  this  special  application,  for  the  use  of  the  U.  S.  Coast 
Survey.     See  Proceedings  of  the  American  Association  for  the 
Adv.  of  Science,  9th  meeting,  p.  97. 

344.  When  an  isolated  observation  of  either  an  immersion  or 
an  emersion  is  to  be  computed,  with  no  corresponding  observa- 
tions at  other  places,  it  will  not  be  necessary  to  compute  the 
values  of  x  and  y  for  a  number  of  hours.     It  will  be  sufficient  to 
compute  them  for  the  time  t  -f  to  (t  being  the   observed  local 
mean  time,  and  to  the  assumed  longitude)  ;  and,  as  the  correction 
of  this  time  will  always  be  small,  the  hourly  changes  may  be 
found  with   sufficient  precision   by  the   approximate   formulae, 
easily  deduced  from  (482), 

da  dd 

x'  =  —  cos  d  y'  =  — 

•K  U  1C 


PREDICTION   FOR   A   GIVEN   PLACE.  557 

where  da  and  dd  denote  the  hourly  increase  of  a  and  d  respect- 
ively. 

345.  To  predict  an  occultation  of  a  given  star  by  the  moon  for  a 
given  place  on  the  earth.  —  We  here  suppose  that  it  is  already  known 
that  the  star  is  to  be  occulted  at  the  given  place  on  a  certain 
date,  and  that  we  wish  to  determine  approximately  the  time  of 
immersion  and  emersion  in  order  to  be  prepared  to  observe  it. 
The  limiting  parallels  of  latitude  between  which  the  occultation 
can  be  observed  will  be  determined  in  the  next  article. 

For  a  precise  computation  we  proceed  by  Art.  322,  making 
the  modifications  indicated  in  Art.  340. 

But,  for  a  sufficient  approximation  in  preparing  for  the  obser- 
vation, the  process  may  be  abridged  by  assuming  that  the  moon's 
right  ascension  and  declination  vary  uniformly  during  the  time 
of  occultation,  and  neglecting  the  small  variation  of  the  parallax. 
It  is  then  no  longer  necessary  to  compute  the  co-ordinates  x  and 
y  directly  for  several  different  times  at  the  first  meridian,  but 
only  for  any  one  assumed  time,  and  then  to  deduce  their  values 
for  any  other  time  by  means  of  their  uniform  changes.  It  will 
be  most  simple  to  find  them  for  the  time  of  true  conjunction  of 
the  moon  and  star  in  right  ascension,  which  is  readily  obtained 
by  the  aid  of  the  hourly  Ephemeris  of  the  moon.  Let  this  time 
be  denoted  by  T0.  We  have  at  this  time  x  —  0,  and  the  value  of 
y  will  be  found  with  sufficient  accuracy  by  the  formula 


in  which  o,  TT,  are  the  moon's  declination  and  horizontal  parallax 
at  the  time  T^  and  df  is  the  star's  declination. 

Let  Aa  (in  seconds  of  arc)  and  A£  here  denote  the  hourly 
changes  of  the  moon's  right  ascension  and  decimation  for  the 
time  T0.  Then  we  have,  nearly, 

Aa  ,         A<$ 

x*  =  —  cos  d  y  '  =  —  • 

7T  7T 

Let  T^  be  any  assumed  time  (which,  in  a  first  approximation, 
may  be  the  time  T0  itself).  Then  the  values  of  the  co-ordinates 
at  this  time  are 


558  OCCULTATIONS    OF   FIXED    STARS. 

and  to  find  the  time  (T)  of  contact  of  the  star  and  the  moon's 
limb,  we  shall,  according  to  Art.  322,.have  the  following  formulae  : 


in  which  /^  is  the  sidereal  time  at  the  first  meridian  corresponding 
to  T19  a'  is  the  star's  right  ascension,  and  co  is  the  longitude  • 

A  sin  B  =  p  sin  <p'  £  =•  p  cos  $>'  sin  # 

A  cos  B  =  p  cos  <p'  cos  #  ^  =  J.  sin  (#  —  <5') 

At'  =  54148  sin  V  Z'=v'A  cos  B 

log  ^'  =9.41916  '  i/  =  A*'  £  sin  *' 

m  sin  M  =  x  —  £  n  sin  N  =  a;'  —  £' 

m  cos  M  =  y  —  ^  n  cos  JV  =  y'  —  if 

m  sin  (M  —  N} 
Bin  4  =  -     7-i  -  log  A  =  9.43500 

A 

A  cos  4       mcos(Jf  —  -ZV) 


where  ^  is  to  be  taken  so  that  cos  ^  shall  be   negative   for 
immersion  and  positive  for  emersion. 

.  For  a  second  approximation,  we  take  T  as  the  assumed  time 
Tv  and  repeat  the  computation  for  immersion  and  emersion 
separately.  The  new  value  of  &  for  this  second  approximation 
will  be  most  readily  found  by  adding  the  sidereal  equivalent  of 
T  (converted  into  arc)  to  its  former  value. 

It  is  more  especially  desirable  to  know  the  true  time  of 
emersion,  and  the  angle  of  position  of  the  point  of  reappearance 
of  the  star.  Since  this  angle  in  solar  eclipses  was  reckoned  on 
the  sun's  limb,  while  here  it  must  be  reckoned  on  the  moon's, 
it  will  be  equal  to  180  -f  Q:  so  that,  taking  the  value  of  $  from 
the  last  approximation,  we  shall  have 

Angle  of  pt.  of  contact  from  the  ")  -  8  ~0    ,     T»T    .     . 

north  pt.  of  thcj  moon's  limb        j  = 

For  the  angle  from  the  vertex  of  the  moon's  limb,  we  find  7-  by 
the  equations 

p  sin  Y  =  £  +  £'T  P  cos  y  —  7  +  VT 


PREDICTION    FOR   A    GIVEN    PLACE.  559 

• 

where  £,  77,  £',  >/,  r  are  to  be  taken  from  the  last  approximation ; 
and  then 

Angle  of  pt.  of  contact  from 
the  vertex  of  the  moon's  limb 

If  the  computation  in  any  case  gives  m  sin  (M—  N)  >  &,  we 
have  the  impossible  value  sin  ^  >  1>  which  shows  that  the  star  is 
not  occulted  at  the  given  place.  If  we  wish  to  know  how  far 
the  star  is  from  the  moon's  limb  at  the  time  of  nearest  approach, 

we  have  (Art.  324) 

A  =  ±  m  sin  (M  —  JV) 

the  sign  being  taken  so  that  A  shall  be  positive.  This  is  the 
linear  distance  of  the  moon's  centre  from  the  line  drawn  from 
the  place  of  observation  to  the  star,  and  therefore  the  angular 
distance  as  seen  from  the  earth  is  xA.  The  apparent  semidiameter 
of  the  moon  is  xk,  and  hence  the  apparent  distance  of  the  star 
from  the  moon's  limb  is  n(J —  &).* 

EXAMPLE. — Find  the  times  of  immersion  and  emersion  in  the 
occultation  of  Aldebaran,  April  15,  1850,  at  Cambridge,  Mass. 

The  elements  of  this  occultation  have  been  found  on  p.  553, 
with  which  an  accurate  computation  may  be  made  by  the 
method  of  Art.  322  ;  but,  according  to  the  preceding  approximate 
method,  we  proceed  as  follows.  The  Greenwich  time  when  the 
moon's  right  ascension  was  equal  to  that  of  the  star  is  found, 
from  the  values  of  a  on  p.  553,  to  be 

T0  =  7*.47  =  7*  28"  12'. 
For  this  time  we  have 

Aa  =  +  2174"  3  =  +  16°  49'  81".l 

A<5  =  +    384  d'=       16    12     1  .7 

n=       3536  *_*'=  +         2249" 

whence,  by  the  above  formulae, 

yQ  =  _|_  0.6360  x'=  +  0.5886  y'  =  +  0.1086 

Then  the  computation  for  Cambridge,  <p  =  42°  22'  49", 
CD  =  4/l  44m  30",  will  be  as  follows.  For  the  first  approximation, 
we  assume  7\  =  7J,  and  hence  we  have 

*  More  exactly,  allowing  for  the  augmentation  of  the  moon's  semidiameter,  it  is 
7T  ( j  _  k)  (1  +  {  sin  TT),  where  we  have  f  =  A  cos  (B  —  6'}. 


560 


OCCULTATIONS   OF   FIXED   STABS. 


T,  =    7*  28™  12'. 

Sid.  time  Gr.  noon  =    1   33      9  .0 

Eeduction  for  T^  =__       1    13 .6 

ftt  =~T    2    34.6 

o'  =    4   27    18.3 

m  =    4  44    30 


—  ./—«  =  *•  =28   50    46.3 
=  357°  41'.6 


with  which  we  find  the  following  results : 

x=      0. 
£  =  _  0.0298 
m  sin  Jf  =  -f  0.0298 
Jlf  =         8°32'.4 

^  =  -+  0.5886 

£'  =  -f  0.1940 

n  sin  JVr  =  -f  0.3946 

N=       74°  19M 
log  sin  4  ==     W9.8395 
mcog(Jf-  JV) 


y  =  -f  0.6360 

,  =  4.  0.4377 

m  cos  M  =  -f-  0.1983 

log  m  =       9.3021 

y>=  +  0.1086 

17'=  — 0.0022 

ncosiV'^  -f-  0.1108 

log  n  =       9.6127 

lOg  COS  4  rrr  9.8590 

*^i  =  :=  0^.4801 


For  immersion. 

T  ~  —    — 

-  0*.6491 

Tl  = 

7  .4700 

T  = 

6  .8209 

T  — 

6*  49-  15« 

to  •= 

4   44    30 

Local  time  = 

2     4    45 

T  = 


For  emersion. 

-  -f  0A.3111 

7  .4700 

=       7  .7811 
7*  46"  52' 
4   44    30 
3     2    22 


Local  time  = 


These  times  are  nearly  correct  enough ;  hut,  for  a  more  accurate 
time  of  emersion,  we  now  assume  Tl=  7*.7811,  with  which  we 
find 

—  T0)  =  +  0.0338 
yt=  +  0.6360 
y  =  +  0,6698 


x  =  x'  (T,  —  T0)  =  +  0.1831 


and"  to  find  the  new  value  of  &  we  have  r  =  -f  0*.3111  =  18m40*, 
the  sidereal  equivalent  of  which  is  18'*  43*.  1,  or  in  arc  4°  40'. 8. 
This,  added  to  the  above  value  of  $,  gives  the  corrected  value 
&  =  2°  22'. 4.  Repeating  the  computation  with  these  new  values 
of  x,  ?/,  and  $,  we  find 


LIMITING    PARALLELS.  561 


7 

N=    74    55 

18° 


=       0.4901 


212    17 


=  —  0.0181 


!T1=        7  .7811  208    34 

T  =       7  .7630 


=      7*45-47* 
Local  time  —      3     1    17 


The  star  reappears  at  212°  17' \ 
from  the  north  point,  or  208°  34'  / 
from  the  vertex,  of  the  moon 'si 
limb.  ) 


This  time  agrees  within  21*  with  the  actually  observed  time  of 
emersion  (given  on  p.  552).  The  principal  part  of  the  difference 
is  due  to  the  error  of  the  Ephemeris  on  this  date. 

346.  To  find  the  limiting  parallels  of  latitude  on  the  earth  for  a 
given  occultation. — The  limiting  curves  within  which  the  occulta- 
tion  of  a  given  star  is  visible  may  be  found  by  the  general 
method  given  for  solar  eclipses,  Art.  311,  which,  of  course,  may 
be  much  abridged  in  such  an  application.  But,  on  account  of 
the  great  number  of  stars  which  may  be  occulted,  it  is  not  pos- 
sible to  make  even  this  abridged  computation  for  all  of  them. 
The  extreme  parallels  of  latitude  are,  however,  found  by  very 
simple  formulae,  and  may  be  used  for  each  star. 

For  a  point  on  the  limiting  curve,  the  least  value  of  J  in  Art. 
324  is  in  a  solar  eclipse  =  £,  but  in  an  occultation  it  is  =  k. 
Hence  we  have,  by  (557),  the  condition 

zb  m  sin  (M  —  N)  =  k 
or,  restoring  the  values  of  m  sin  M=  x  —  £ ,  m  cos  M=  y  —  y, 

(x  —  c)  cos  N  —  (y  —  77)  sin  N  =  ±  k 

The  angle  Nis  here  determined  by  the  equations  (552);  but,  for 
an  approximate  determination  of  the  limits  quite  sufficient  for 
our  present  purpose,  we  may  neglect  the  changes  of  £  and  7,  and 
take 

n  sin  N  =  xf  n  cos  N  =  y' 

Let  x0  and  ?/0  be  the  values  of  x  and  y  for  the  assumed  epoch 
TQ ;  then  for  any  time  T=  T0  -j-  r  we  have 

x  =  X0  -f-  n  siix  N  .  r  y  =  y0  -j-  n  cos  N .  r 

VOL.  I.— 36 


562  OCCULTATIONS  OF  FIXED  STARS. 

which  reduce  the  above  condition  to 

(XQ  —  £)  cos  N  —  (y0  —  ij)  sin  N  =  ±  k 

By  the  last  equation  of  (500),  we  have,  by  neglecting  the  com- 
pression of  the  earth, 

sin  y  =  TI  cos  d'  -}-  C  sin  3' 
in  which 


and  we  are  now  to  determine  the  maximum  and  mininum  values 
of  ^>,  which  fulfil  these  conditions.     Let  us  put 

a  =  —  £  cos  N  -f-  ^  sin  -#" 
6  =       £  sin  JVr  -f-  ^  cos  N 

from  which  follow 

.     £  =  —  a  cos  .ZV  -f  6  sin  N  ' 
7]  —       a  sin  JV  -j-  6  cos  JV 
C  -        -/(I  —  a2  —  &2) 

Then  we  also  have,  by  our  first  condition, 

a  =  —  #0  cos  JV  -f  y0  sin  JV  ±  A 

which  is  a  constant  quantity,  since  we  may  here  assume  x'  and  yf 
to  be  constant. 

Since  we  have  a2-}-  62-f  <^—  1,  we  can  assume  p  and  e  so  as  to 

satisfy  the  equations 

cos  f  =  a 
sin  f  cos  e  =  b 
sin  Y  sin  e  =  C 

in  which  sin  y  may  be  restricted  to  positive  values.  -The  formula 
for  (p  thus  becomes 

sin  <p  =  cos  f  sin  N  cos  3'  -j-  sin  y  cos  e  cos  JV"  cos  <S'  -f  sin  y  sin  e  8in  ^' 


which  may  be  put  under  a  more  simple  form  by  assuming  /?  and 
^,  so  as  to  satisfy  the  conditions 

sin  /9  =  sin  N  cos  5' 
cos  /5  cos  A  =  cos  JV  cos  df 
cos  /?  sin  A  =  sin  5' 

in  which  cos  /9  may  be  restricted  to  positive  values. 


LIMITING    PARALLELS.  563 

We  thus  obtain 

sin  <f>  =  sin  ft  cos  f  -j-  cos  /9  sin  y  cos  (A  —  e) 

in  which  ^  and  e  are  the  only  variables.  Since  cos  ft  sin  f  is 
positive,  this  value  of  sin  <p  is  a  maximum  when  cos  (k  —  e)  =  1 
or  A  —  £  =  0  ;  and  a  minimum  when  cos  (A  —  e)  =  —  1,  or 
^  —  e  —  180°.  Hence  we  have,  for  the  limits,  sin  <p  =  sm(ft  -±  7-), 

that  is 

for  the  northern  limit,     <p  =  ft  -(-  f 
for  the  southern  limit,     <f>  =  /S  —  y 

One  of  the  points  thus  determined  may,  however,  be  upon 
that  side  of  the  earth  which  is  farthest  from  the  moon,  since  we 
have  not  restricted  the  sign  of  £,  and  our  general  equations 
express  the  condition  that  the  point  of  observation  lies  in  a  line 
drawn  from  the  star  tangent  to  the  moon's  limb,  which  line 
intersects  the  surface  of  the  earth  in  two  points,  for  one  of  which 
£  is  positive  and  for  the  other  negative.  But,  taking  £  only  with 
the  positive  sign,  we  must  also  have  sin  e  positive.  For  the 
northern  limit,  therefore,  when  ^  =  e,  sin  A  must  be  positive, 
which,  according  to  the  equation  cos  ft  sin  ^  =  sin  £',  can  be  the 
case  only  when  S'  is  positive.  Hence  the  formula  y>  =  ft  -}-  f 
gives  the  most  northern  limit  of  visibility  only  when  the  star  is 
in  north  declination.  For  similar  reasons,  the  formula  <p=ft —  f 
gives  the  southern  limit  only  when  the  star  is  in  south  declina- 
tion. The  second  limit  of  visibility  in  each  case  must  evidently 
be  one  of  the  points  in  which  the  general  northern  or  southern 
limiting  curve  meets  the  rising  and  setting  limits, — namely,  the 
points  where  £  =  0,  and  consequently,  also,  sin  e  =  0,  cos  e  =  ±  1, 
which  conditions  reduce  the  general  formula  for  sin  tp  to  the 
following : 

sin  <p  =  (sin  N  cos  f  ±  cos  N  sin  y)  cos  8'  =  sin  (N  ±  y)  cos  3f 

If  cos  N  is  taken  with  the  positive  sign  only,  the  upper  sign  in 
this  equation  will  give  the  most  northern  limit  to  be  used  when 
the  southern  limit  has  been  found  by  the  formula  tp  =  ft  —  f\  and 
the  lower  sign  will  give  the  southern  limit  to  be  used  when  the 
northern  limit  has  been  found  by  the  formula  <p  =  ft  -f  f. 

Finally,  since  the  epoch  T0  is  arbitrary,  we  may  assume  for  it 
the  time  of  true  conjunction  in  right  ascension  when  x0—  0,  and 
we  shall  then  have 

a  =  cos  r  =  y0s\uN  ±  k 


564  OCCULTATIONS    OF    FIXED    STARS. 

The  above  discussion  leads  to  the  following  simple  arrangement 
of  the  formulae 

cos  ft  =  y0  sin  N  ±  0.2723  (r  <  180°) 

sin  y9  =  sin  NCOS  dr  .     (£<    90°) 

(604) 

cos  ft  =t  y0sin  N  q=  0.2723 
sin  <?2=  sin  (JVrq=  ya)  cos  ^5r  (-ZV<    90°) 


in  which  the  upper  or  the  lower  signs  are  to  be  used,  according 
as  the  declination  of  the  star  is  north  or  south.  When  the 
declination  is  north,  ^  will  be  the  northern  limit  and  <p2  the 
southern ;  and  the  reverse  when  the  declination  is  south.  The 
angle  7VTis  here  supposed  to  be  less  than  90°,  and  is  found  by 
the  formula 

tan  ^=^ 

y 

always  considering  yf  as  well  as  x'  to  be  positive. 

When  the  cylindrical  shadow  extends  beyond  the  earth,  north 
or  south,  we  shall  obtain  imaginary  values  for  ft  or  Tz-  The 
following  obvious  precepts  must  then  be  observed : 

1st.  When  cos  ft  is  imaginary,  the  occupation  is  visible  beyond 
the  pole  which  is  elevated  above  the  principal  plane  of  reference, 
and,  therefore,  we  must  put  for  the  extreme  limit  <pl=-  -f-  90°,  or 
(pl=  —  90°,  according  to  the  sign  of  3'. 

2d.  When  cos  ^  is  imaginary,  the  value  of  <p2  will  be  the  lati- 
tude of  that  point  of  the  (great  circle)  intersection  of  the  prin- 
cipal plane  and  the  earth's  surface  which  lies  nearest  the  depressed 
pole;  that  is,  we  must  take  <p2=  d' —  90°,  or  ^,=  ^+90°, 
according  as  3f  is  positive  or  negative. 

It  is  also  to  be  observed  that  the  numerical  value  of  <pl 
obtained  by  the  formula  <pl=  ft  ±  ft  may  exceed  90°,  in  which 
case  the  true  value  is  either  <p^=  180°--  (/9  zb  ft),  or  ^^—  -  180° 
-  (/9  ±:  ft),  since  these  values  have  the  same  sine. 

EXAMPLE. — Find  the  limiting  parallels  of  latitude  for  the 
occultation  of  Aldebaran,  April  15,  1850. 

We  have  found,  page  559,  for  this  occultation, 

y0  =  -f  0.6360  x'  m  0.5886  yf  =  0.1086 

Hence,  with  S'=  16°  12r,  we  find 


OCCULTATIONS    OF    PLANETS.  56«5 

N  =       79°  33'  log  sin  ,3  =  9.9751 

y0  sin  N  =  +  0.6255  /?  a±  70°  47' 

A-  =t       0.2723  ri  =  26      8 

cos  ft  =  -f  0.8978  ft  -f  ft  =  96    55 

cos  ft  =  +  0.3532  ^  =  83      5 

ft  =  69°    19' 

N—r*=         10    14  <pz=    9.  49 

It  is  hardly  necessary  to  observe  that  the  occultation  is  not 
visible  at  all  the  places  included  between  the  extreme  latitudes 
thus  found,  since  the  true  limiting  curves  do  not  coincide  with 
the  parallels  of  latitude,  but  cut  the  meridians  at  various  angles, 
as  is  illustrated  by  the  southern  limit  in  our  diagram  of  a  solar 
eclipse,  p.  504.  Unless  a  place  is  considerably  within  the 
assigned  limits,  it  may,  therefore,  be  necessary  in  many  cases  to 
make  a  special  computation,  by  the  method  of  Art.  345,  to  deter- 
mine whether  the  occultation  can  be  observed. 


OCCULTATIONS    OF    PLANETS    BY    THE    MOON. 

347.  If  the  disc  of  a  planet  were  always  a  circle,  and  fully 
illuminated,  its  occultation  by  the  moon  might  be  computed  by 
the  general  method  used  for  solar  eclipses  by  merely  substituting 
the  parallax  and  semidiameter  of  the  planet  for  those  of  the  sun  ; 
and  this  is  the  method  which  has  generally  been  prescribed  by 
writers  on  this  subject.  But  with  the  telescopes  now  in  use, 
and  especially  with  the  aid  of  the  electro-chronograph,  it  is 
possible  to  observe  the  instants  of  contact  with  the  planet's  limb 
to  such  a  degree  of  accuracy  that  it  appears  to  be  worth  while 
to  take  into  account  the  true  figure  of  the  visible  illuminated 
portion  of  the  planet.  Moreover,  the  investigation  of  this  true 
figure  possesses  an  intrinsic  interest  which  justifies  entering  upon 
it  here  somewhat  at  length. 

In  order  to  embrace  at  once  all  cases,  I  shall  consider  the 
planet  as  a  spheroidal  body  which  even  when  fully  illumi- 
nated presents  an  elliptical  outline,  and  when  partially  illumi- 
nated presents  an  outline  composed  of  two  ellipses,  of  which 
one  is  the  boundary  of  the  spheroid  and  the  other  is  the  limit  of 
illumination  on  the  side  of  the  planet  towards  the  observer.  I 
begin  with  the  determination  of  the  first  of  these  ellipses. 


OCCULTATIONS    OF    PLANETS. 

848.  To  find  the  apparent  form  of  the  disc  of  a  spheroidal  planet.*  — 
Let  us  first  express  the  apparent  place  of  any  point  of  the 
surface  of  the  planet,  by  referring  it  to  three  planes  perpen- 
dicular to  each  other,  of  which  the  plane  of  xy  coincides  with  the 
plane  of  the  planet's  equator,  while  the  axis  of  z  coincides  with 
the  axis  of  rotation.  In  this  system,  let 

x,y,z  =  the  co-ordinates  of  the  point  on  the  surface  of  the 

planet, 
£,  T),  C  =  those  of  the  observer. 

Straight  lines  drawn  from  the  observer  to  the  centre  of  the 
planet  and  to  the  point  on  its  surface  determine  their  apparent 
places  on  the  celestial  sphere.  If  these  places  are  referred  to 
the  great  circle  which  corresponds  to  the  planet's  equator,  and 
if  we  put 

/I,  X'  =.  the  geocentric  longitudes  of  tl^e  apparent  places  of  the 
planet's  centre  and  the  point  on  its  surface,  reckoned 
from  the  axis  of  x,  in  the  great  circle  of  the  planet's 
equator, 

£,/9'=the  latitudes  of  these  places  referred  to  the  great 
circle  of  the  planet's  equator, 

/>,  p'  =  the  distances  of  the  centre  of  the  planet  and  the  point 
on  its  surface  from  the  observer, 

we  shall  have  (Arts.  32  and  33)f 

p  cos  p  cos  X=     -  $  ~\ 

P  cos  p  sin  A  =  —  7]  V    (605) 

p  sin  j3  =  —  C  J 


cos  ft  cos 
'          ' 


cos/5'  sin  /.'  =  y  —  y  V    (606) 

p'  sin  /5'  =  2  —  C  J 


*  The  method  of  investigation  here  adopted,  so  far  as  relates  to  the  apparent  form  of 
the  disc,  is  chiefly  derived  from  BESSEL,  Astronomische  ITntersuchungen,  Vol.  I.  Art.  VI. 

|  The  group  (606)  may  be  deduced  by  supposing  for  a  moment  that  the  position 
of  the  observer  is  referred  to  a  system  of  planes  parallel  to  the  first,  but  having  its 
origin  at  the  point  on  the  surface  of  the  planet.  The  co-ordinates  in  this  system  are 
equal  to  those  in  the  first  increased  respectively  by  x,  y,  and  z.  The  negative  sign 
in  the  second  members  of  both  groups  results  from  the  consideration  that  the  longi- 
tude of  the  observer  as  seen  from  the  planet  is  180°  -j-  \  or  180°  -\-  A';  and  his 
latitude,  —  ft,  or  —  ft'.  Compare  Art.  98. 


FORM   OF   A   PLANETS   DISC. 


567 


Now,  let  0  and  C,  Fig.  47,  be  the  apparent          Fig.  4r. 
places  of  the  planet's  centre  and  the  point  on  its 
surface,  projected  upon  the  celestial  sphere;   Q 
the  pole  of  the  planet's  equator;  P  the  pole  of  the 
earth's  equator ;  and  let 

s'  =  the  apparent  distance  of  C  from  0  =  the  arc 

00, 
p'=  the  position  angle  of  C  reckoned  at  0,  from 

the  declination  circle  OP  towards  the  east, 

=  POC, 
p=  the  position  angle  of  the  pole  of  the  planet 

=  POQ; 

then,  in  the  triangle  QOC,  we  have 

sin  s'  sin  (p'  —  jp)  =  cos  0'  sin  (A'  —  A) 

sin  sf  cos  (pr  —  p)  =  cos  0  sin  0' —  sin  0  cos  0'  cos  (A'  —  A) 

Multiplying  these  by  p',  and  substituting  the  expressions  (605) 
and  (606),  we  obtain 

p'  sin  s'  sin  (p'  —  p)  =  —  x  sin  A  -f  y  cos  A 

p'  sin  s'  cos  (pf  —  p)  =  —  x  sin  0  cos  A  —  y  sin  0  sin  A  -f  z  cos  0 

or,  since  s'  is  very  small  and  p'  sin  sf  or  p's'  differs  insensibly 
from  p  sin  sf  or  psf, 

ps'  sin  (/  —  p)  =  —  x  sin  A  -f  y  cos  A  ) 

/>s'  cos  (p'  —  p)  =  —  #  sin  /?  cos  A  —  y  sin  0  sin  A  -j-  2  cos  0  ) 

These  equations  apply  to  any  point  on  the  surface  of  the  planet. 
If  we  apply  them  to  those  points  in  which  the  visual  line  of  the 
observer  is  tangent  to  that  surface,  they  will  determine  the  curve 
which  forms  the  apparent  disc.  The  equation  of  an  ellipsoid  of 
revolution  whose  axes  are  a  and  b,  of  which  b  is  the  axis  of 
revolution,  is 

""  +  r!  (608) 


aa 


aa 


bb 


and  the  equation  of  a  tangent  line  passing  through  the  point 
whose  co-ordinates  are  f ,  37,  and  £  is 


i  =    JL    4. 

aa       aa       bb 


(609) 


The  distances  f ,  37,  and  f  are  very  great  in  comparison  with  x, 


568  OCCULTATIONS    OF    PLANETS. 

t.  ** 

y,  and  z.     If  we  divide  (609)  by  /?,  the  quotients  ~>  -,  -  will  be 

of  the  same  order  as  -,  -,  r,  but  the  quotient  -  will  be  inappre- 

T*       ?y        2 

ciable  in  relation  to  the  quotients  — ,  — ,  -77-*     Performing  this 

aa    aa     bb 

division,  therefore,  and  substituting  the  values  of  £,  37,  and  £  from 
(605),  we  may  write  for  the  equation  of  the  tangent  line 

x  cos  ft  cos  A       y  cos  /5  sin  X       z  sin  /5 

-55-  -55-     +  ~&T- 

If  the  curve  ACB,  Fig.  47,  is  referred  to  rectangular  axes 
passing  through  the  apparent  centre  0  of  the  planet,  one  of 
which  is  in  the  direction  of  the  pole  of  the  planet,  and  if  u  and 
v  denote  the  co-ordinates  of  any  point  of  the  curve,  so  that 

u  =  s'  sin  (pf  —  p) 
v  =  s'  cos  (pr  —  p) 

the  equations  (607)  and  (610)  will  enable  us  to  determine  x,  y, 
and  z  in  terms  of  u  and  v.     Putting 

bb 

—  =  1  —  ee 

aa 

the  three  equations  become 

pu  =  —  x  sin  A  -j-  y  cos  /I 
pv  =  —  (x  cos  /  -f  y  sin  A)  sin  /5  -j-  z  cos  /? 
0  =  (x  cos  A  -|-  ?/  sin  A)  (1  —  ee)  cos  /3  -J-  2  sin  /3 

from  which  we  derive 

* 
—  x  sin  A  -f-  y  cos  k  =  pu 

sin 
—  #  cos  A  —  y  &\n  I  =  pv 


z  =  pv 


1  —  ee  cos2/5 
(1  —  ee}  cos 


1  —  ee  cos2/? 
Substituting  these  values  in  (608)  and  putting 

s  =  —  =  the  greatest  apparent  semidiameter  of  the  planet, 
ee  cos2  /9) 


FORM  OF  A  PLANET'S  DISC.  569 

we  find 

vv 
ss  =  uu  +  -  (611) 

which  is  the  equation  of  the  outline  of  the  planet  as  projected 
upon  the  celestial  sphere,  or  upon  a  plane  passed  through  the 
centre  of  the  planet  at  right  angles  to  the  line  of  vision.  It 
represents  an  ellipse  whose  axes  are  2 s  and  2s  ]/(!  —  ee  cos2/9), 
e  heing  the  eccentricity  of  the  planet's  meridians.  The  minor 
axis  (OB,  Fig.  47)  lies  in  the  direction  of  the  great  circle  drawn 
to  the  pole  of  the  planet's  equator. 

We  next  proceed  to  determine  what  portion  of  this  ellipse  is 
illuminated  and  visible  from  the  earth. 

349.  To  find  the  apparent  curve  of  illumination  of  a  planeCs  surface. — 
If  the  sun  be  regarded  as  a  point  (which  will  produce  no  sensible 
error  in  this  problem),  the  curve  of  illumination  of  the  planet,  as 
seen  from  the  sun,  can  be  determined  by  conditions  quite  similar 
to  those  employed  in  the  preceding  problem ;  for  we  have  only 
to  substitute  the  co-ordinates  expressing  the  sun's  position  with 
reference  to  the  planet,  instead  of  those  of  the  observer.  If, 
therefore,  we  put 

Ay  B  =  the  heliocentric  longitude  and  latitude  of  the  centre 
of  the  planet  referred  to  the  great  circle  of  the 
planet's  equator, 

the  equation  of  the  tangent  line  from  the  sun  to  the  planet, 
being  of  the  same  form  as  (610),  will  be 

,  x  cos  B  cos  A         y  cos  5  sin  yl       z  sin  5 

~~c^T  ~~CLO~  ~W~ 

If  each  point  which  satisfies  this  condition  be  projected  upon 
the  celestial  sphere  by  a  line  from  the  observer  on  the  earth,  and 
u  and  v  again  denote  the  co-ordinates  of  the  projected  curve,  we 
have  here,  also,  to  satisfy  the  equations 

pu  =  —  x  sin  A  -|-  y  cos  A  ^ 

pv  =  —  (x  cos  /I  -f  y  sin  A)  sin  /9  -|-  z  cos  /5  / 

in  which  ^  and  ft  have  the  same  signification  as  in  the  preceding 
article.  The  values  of  x,  y,  and  z,  determined  by  the  three 
equations  (612),  (613),  being  substituted  in  the  equation  of  the 
ellipsoid,  we  obtain  the  relation  between  u  and  v,  or  the  equation 


570  OCCULTATIONS  OF  PLANETS. 

of  the  required  curve  of  illumination  as  seen  from  the  earth.    In 
order  to  facilitate  the  substitution,  let  us  put 

xl=  —  x  sin  A  -f-  y  cos  A 
yl  =       x  cos  A  -j-  y  sin  A 
from  which  follow 

x=  —  xl  sin  A  -f-  yl  cos  A 
y  =        xl  cos  A  -f-  i/j  sin  A 

At  the  same  time,  let  us  introduce  the   auxiliaries  ft  and  J3t 
dependent  upon  /9  and  1?  by  the  assumed  relations 


-  cos  ft  —  cos  /5  —  cos  .Bj  =  cos  J5 

-  sin  ft  =  -  sin  ft  —  sin Bl=  -  sin  1? 
go  (JT  o 


(614) 


Then  the  three  equations  become 

0  =       #,  cos  .Sj  sin  (yl  —  A)  -f  yl  cos  ^  cos  (yl  —  A)  -j-  -  z  sin  £, 
/>w  =       #! 

£$?»  =     ~  2/1  sin  PI  +  I*  cos  ^ 
from  which  we  derive 

xl=      f>u 
Nyl  =i  —  pu  cos  /5j  cos  Sl  sin  (^1  —  A)  —  -  gpvsinBl 

N  -  2  =  —  pu  sin  ft  cos  ^sin  (A  —  A)  -\-7gpv  cos  J^cos^  —  A) 

where,  for  brevity,  jYisput  for  sin  ft  sin  B^  -\-  cos  ft  cos  jBj  cos  (A — A). 
Before  substituting  these  expressions  in  the  equation  of  the 
ellipsoid,  it  will  be  well  to  consider  the  geometrical  signification 
of  the  quantities  ft  and  Br  If  we  draw  straight  lines  from  the 
centre  of  the  planet  to  the  earth  and  to  the  sun,  the  latitudes  of 
the  points  in  which  these  lines  intersect  the  surface  of  the  planet 
will  be  ft  and  B.  If  these  points  be  projected  upon  the  surface 
of  a  sphere  circumscribed  about  the  ellipsoid,  by  perpendiculars 
to  its  equator,  the  latitudes  of  the  projected  points  will  be  ft  and 
BI',  and$  and  G  will  be  the  corresponding  radii  of  the  ellipsoid. 
If  now  these  projected  points  are  referred  to  the  celestial  sphere, 
by  lines  from  the  planet's  centre,  they  will  form  with  the  pole  Q 
of  the  planet's  equator  a  spherical  triangle  QOS,  in  which  the 


FORM  OF  A  PLANET'S  DISC.  571 

angle  Q  will  be  A  —  X  ;  and  the  sides  including  this  angle  wilJ 
be  90°  —  ft  =  §0,  90°  —  Bl  =  QS.  Denoting  the  angle  at  0  by 
w,  and  the  side  0$  by  F,  we  shall  have 

cos  V  =  sin  ft  sin  Bl  -f  cos  ft  cos  B^  cos  (A  —  /I)  ^| 
sin  V  cos  w;  =  cos  ft  sin  5,  —  sin  ft  cos  J5A  cos  (A  —  A)  >    (615) 
sin  V  sin  w  =  cos  Bl  sin  (/I  —  A)  ) 

in  which  V  is  very  nearly  the  angular  distance  between  the  sun 
and  the  earth  as  seen  from  the  planet. 
This  triangle  also  gives 

sin  B^  =  cos  Fsin  ft  -f  sin  Fcos  ft  cos  w 
cos  B1  cos  (A  —  A)  =  cos  V  cos  ft  —  sin  V  sin  ft  cos  w 
cos  Bt  sin  (A  —  X)  =  sin  Fsin  w 

By  these  equations  the  above  expressions  for  xv  ylt  and  z  are 
reduced  to 

cos  V.  xt  =      pu  cos  V 

cos  V  .yl  =  —  pu  sin  F  sin  w  cos  ft 

-  -  gpv  (cos  F  sin  ft  -}-  sin  F  cos  ft  cos  w) 
cos  V--Z  —  —  pu  sin  F  sin  w  sin  ft 

+  I  9PV  (cos  "P"  cos  ^i  —  s^n  ^  sin  ft  cos  w) 

Substituting  these  in  (608),  observing  that  xx  -\-  yy  =  x^  +  y^v 
we  have 

COs2F—  =  wwcos2F 
PP 

-f-    (w  sin  w;  -f-  -  gv  cos  M?)  sin  F  cos  ft  -f  -  gv  cos  F  sin  ft  j 
+    (w  sin  u;  -{-  -  </y  cos  w)  sin  F  sin  ft  —  .-  £u  cos  F  cos  ft 

Developing  the  squares  in  the  second  member,  and  putting  5  for 
-,  and  also 

c  =  -|/(1  —  ee  cos2  ft)  =  - 
we  shall  find 


ss 


=  i  u  cos  w  —  v-—    -l-flwsinitf-fi;.—     -|  sec2  F    (616) 


572 


OCCULTATIONS    OF    PLANETS. 


which  is  the  required  equation  of  the  curve  of  illumination,  as 
seen  from  the  earth,  projected  upon  the  celestial  sphere.  It 
represents  an  ellipse  whose  centre  is  at  the  origin  but  whose 
axes  are,  in  general,  inclined  to  the  axes  of  co-ordinates,  and, 
consequently,  to  the  axes  of  the  ellipse  of  equation  (611).  The 
equation  (611)  is  only  the  particular  case  of  (616)  which  corre- 
sponds to  V=  0,  or  the  case  of  full  illumination. 


Fig.  48. 


350.  We  have  yet  to  determine  what  portions  of  the  apparent 

disc  are  bounded  by  the  two  curves 
respectively.  If  ABA'B',  Fig.  48, 
is  the  ellipse  of  (611),  which  I  shall 
call  ihe  first  ellipse,  and  CDC'Df  that 
of  (616),  which  I  shall  distinguish  as 
the  second  ellipse,  the  visible  outline 
of  the  planet  is  composed  of  one- 
half  the  first  and  one-half  the  second 
curve,  and  these  halves  either  begin  or  end  at  the  points  C  and 
C",  which  are  the  common  points  of  tangency  of  the  two  curves. 
These  points  satisfy  both  equations;  and,  therefore,  putting  uv  and 
I-!  for  the  co-ordinates  of  either  point,  and  subtracting  (611)  from 

(616),  we  find 

/  cos  w  \2  ,     „  Tr  • 

0  =  I  w,  sin  w  -f  vl  -  J  tan2  I 

which  is  satisfied,  in  general,  by  taking 

cos  w       n 
w  sin  w  -\-  vl  - 

C 

Denoting  the  position  angle  corresponding  to  u^  vv  by  p»  we 
have  ul=  sl  sm(pl—  p),  vl  =  sl  cos^  —  p).  Substituting  these 
values,  and  also  putting 


c1  sin  wl  =  sin  w  ^  cos  wt  — 

the  preceding  condition  becomes 

c^cosQ^  —  p  —  u\)  =  0 
whence 


cos  w 


(617) 


(618) 


which  expresses  the  position  angles  of  both  C  and  C'.     If  we 
draw  the  arc  ODO',  Fig.  48,  making  the  angle  B00f  =  M?,  and 


FORM  OF  A  PLANET'S  DISC.  573 

take  00 '  =  V,  the  point  0'  will  be  nearly  the  position  of  the 
planet  as  seen  from  the  sun,  and  the  arc  Kwill  be  the  measure 
of  the  angular  distance  between  the  sun  and  the  earth  as  viewed 
from  the  planet.  If  we  assume  sin  w  to  be  positive  in  equations 
(615),  as  \ve  are  at  liberty  to  do,  the  arc  "Fwill  be  reckoned  from 
the  planet  eastward  from  0°  to  360°.  Now,  so  long  asT^is  less 
than  180°,  the  west  limb  will  evidently  be  the  full  limb,  and 
when  Vis  greater  than  180°,  the  east  limb  will  be  the  full  limb. 
Hence  we  infer  that  a  point  whose  given  position  angle  is  p'  is 
on  the  east  limb  when 

p'  >  p  -f  wl  —  90°     and     <  p  -f  wl  -f  90° 
but  on  the  west  limb  when 

p'  <  p  -f  wl  —  90°     and     >  p  -f  wt  -f  90° 

When  V  >  90°  and  <  270°,  the  planet  is  crescent ;  but  when 
F>  270°  and  <  90°,  it  is  gibbous.  In  the  case  of  a  crescent 
planet  there  are  two  points,  one  on  the  full  and  the  other  011  the 
crescent  limb,  corresponding  to  the  same  position  angle :  hence 
in  observations  of  a  crescent  planet  the  point  of  observation  on 
the  limb  will  not  be  sufficiently  determined  by  the  position 
angle  alone  ;  it  will  be  necessary  for  the  observer  to  distinguish 
the  crescent  from  the  full  limb  in  his  record. 

351.  In  order  to  apply  the  preceding  theory,  it  is  necessary  to 
find  the  quantities  p,  ft,  ^  B,  A.  The  direction  of  the  axis  of  x 
in  Art.  348  was  left  indeterminate,  and  may  be  assumed  at 
pleasure,  but  it  is  most  convenient  to  let  it  pass  through  the 
ascending  node  of  the  planet's  equator  on  the  equinoctial,  so  that 
/  and  A  will  be  reckoned  from  this  node.  The  position  of  the 
node  must,  therefore,  be  known,  and  this  we  derive  from  the 
researches  of  physical  astronomers.  If  we  put 

n  =  the  longitude  of  the  ascending  node  of  the  planet's 
equator  on  the  equinoctial, 

i  =  the  inclination  of  the  planet's  equator  to  the  equi- 
noctial, 

we  have  at  any  given  time  /,  for  the  planets  Jupiter  and  Saturn, 
the  only  ones  whose  figures  are  sensibly  spheroidal, 


574  OCCULTATIONS    OF    PLANETS. 

For  Jupiter     {  n  =  357°  56'  25"  +  3"'59  «  ~  185°) 
I  i  =    25°  25'  49"  +  0".66  (t  —  1850) 

ForS        n   f  w  =  125°13'54"  +  128".7G(^—  1850)  +  0".0605(^  —  1850)2 
'\  i=     7°  10'  10"—   15".08(<  —  1850)  +  0".0035(£  —  1850)2 


in  which  t  is  expressed  in  years.* 

The  values  for  Saturn  apply  either  to  its  equator  or  the  rings, 
which  are  sensibly  in  the  same  plane. 

If  now  we  put 

a',  dr  =  the  right  ascension  and  declination  of  the  planet, 

we  can  convert  a'  and  3f  into  A  and  /?  by  Art.  23  ;  we  shall 
merely  have  to  substitute  in  (29)  or  (31)  a'  —  n  for  a,  dr  for  $, 
and  i  for  e.  The  angle  p  is  here  the  position  angle  of  the  pole 
of  the  planet  reckoned  from  the  declination  circle  of  the  planet 
towards  the  east;  but  in  Art.  25  the  angle  y  is  the  position 
angle  reckoned  towards  the  west,  and,  therefore,  we  shall  have 
to  put  y  =  360°  —  p  in  (33).  Hence  we  obtain  the  following 
formulae  for  fa  A,  and  p  : 


/  sin  F  =  tan  dr  f  sin  /I  =  cos  (F  —  i) 

f  cosF  =  sin  (a'  —  ri)          .       /'  cos  A  =  cos  F  cot  (a'  —  n) 
tan  /5  =  sin  A  tan  ( JF1  —  j) 

sin  .F'  cot  (a'  —  w) 
tan  JP'=  tan  i  sin  (a'  —  n)         tan  p  =  — 


cos        — 


(619) 


To  find  A  and  B,  we  avail  ourselves  of  the  heliocentric  longi- 
tude and  latitude  of  the  planets  given  in  the  British  Almanac, 
and  as  these  quantities  are  referred  to  the  ecliptic,  while  A  and 
B  are  referred  to  the  planet's  equator,  we  must  know  the  rela- 
tive position  of  these  circles.  Putting 

N' =  the  longitude  of  the  node  of  the  planet's  equator  on 

the  ecliptic, 

/'=  the  inclination  of  the  planet's  equator  to  the  ecliptiCj 
.TV— the   arc   of  the   planet's   equator   between    the  equi- 
noctial and  the  ecliptic, 

*  These  values  I  have  deduced  from  the  data  given  in  DAMOISEAU'S  Tables  Eclip- 
tiques  des  Satellites  de  Jupiter,  Paris,  1836 ;  and  BESSEL'S  Bestimmung  der  Lage  und 
Grbsse  des  Satnrns-Ringes  und  der  Figur  und  Grosse  des  Saturns,  Astronorn.  Nach,,  Vol. 
XII.  p.  167. 


FORM  OF  A  PLANET'S  DISC.  575 

we  deduce  from  the  data  of  BESSEL  and  DAMOISEAU,  for  a  given 
year  t, 

r  N'=  335°  40'  46"+  49".80  (t  — 1850) 
For  Jupiter.  J    I'=     2°   8'51"+   0".43(£  — 1850) 

(  N  =  336°  33'  18"+  46".55  (t  — 1850) 

(  N'=  167°  31'  52"+  46".62  (t  — 1850) 
For  Saturn.  J    /'=  28°  10' 27"—  0".35(f  — 1850) 

(N=  43°  31'  34"—  86".75  (t  — 1850)  —  0".0625(*  — 1850)2 

and  these  values  for  Saturn  also  apply  to  the  rings. 
Finally,  if  we  put 

A'j  .B'=the    heliocentric    longitude    and    latitude   of   the 
planet,  referred  to  the  ecliptic, 

the  formulae  (29)  or  (31)  will  serve  to  convert  A'  —  N1  and  Bf 
into  A  —  N  and  B  ;  and  they  become 


K  sin  M  =  tan  B'  Kf  sin  (A—N)  =  cos  (M  —  /') 

K  cos  M  =  sin  (A'—Nr)       K'  cos  (A—N)  =  cos  M  cot  (A'—N') 

tan  B  =  sin  (A  —  N)  tan  (M  —  /') 


(620) 


352.  The  preceding  complete  theory  admits  of  several  abridg- 
ments in  its  application  to  the  different  planets,  varying  according 
to  the  features  peculiar  to  each. 

Jupiter. — The  inclination  of  Jupiter's  equator  to  the  ecliptic  is 
so  small  that  the  quantity  <?=j/(l  —  eecos2/?)  never  differs 
sensibly  from  |/(1  —  ee\  which,  according  to  STRUVE'S  measures, 
is  0.92723.  I  shall,  therefore,  use  as  a  constant  the  value 
log  c  =  9.9672.  Again,  on  account  of  the  small  inclinations  both 
of  Jupiter's  equator  and  of  his  orbit  to  the  ecliptic,  the  angle  w 
never  differs  much  from  90°,  and,  since  this  angle  is  required 
only  in  computing  the  gibbosity  of  the  planet  (which  never 
exceeds  0".5),  it  is  plain  that  we  may  take  w  =  90°,  and  that  V 
may  be  found  with  sufficient  accuracy  by  the  formula 

V=  A  —  X 
or,  indeed,  by  the  formula 

V=A'  —  X  (621) 

in  which  A'  and  X  are,  respectively,  the  heliocentric  and  geo- 
centric longitudes  of  the  planet,  the  former  being  taken  directly 


576  OCCULTATIONS   OP   PLANETS. 

from  the  British  Almanac,  and  the  latter  computed  from  the  geo- 
centric right  ascension  and  declination  by  Art.  23 :  so  that  for 
this  planet  the  equations  (615),  (619),  (620)  will  be  dispensed 
with,  except  only  the  last  two  equations  of  (619),  which  will  be 
required  in  finding  p. 

Saturn. — The  inclination  of  Saturn's  equator  to  the  ecliptic  is 
over  28°,  and  therefore  the  quantity  c  =  |/(1  —  ee  cos2  /9)  will 
have  sensibly  different  values  at  different  times.  The  value  of 
—  /9  is,  however,  given  in  the  table  for  Saturn's  Ring  in  our 
Ephemerides  (where  it  is  usually  denoted  by  1).  The  value  of  ce 
is  0.1865,  or  log  ee  =  9.2T06.  The  gibbosity  of  Saturn  is  alto- 
gether insensible ;  so  that  we  shall  have  occasion  to  use  only  the 
equation  (611),  or  in  any  formula  that  may  be  derived  from  the 
more  general  equation  (616)  we  shall  have  to  put  V=  0.  The 
angle  p  is  also  given  in  the  table  for  the  ring. 

Saturn's  Ring. — The  ring  may  be  here  regarded  as  an  ellipsoid 
of  revolution  whose  minor  axis  —  0.  Hence  we  have  only  to 
make  e  =  1  in  our  formulae  to  obtain  the  equation  of  its  elliptical 
outline.  This  gives  c  =  y{\  —  cos2/?)  =  sin/9,  which  value  being 
substituted  in  (611),  we  have  at  once  the  required  equation, 
while  the  position  of  the  ellipse  is  given  at  once  by  the  angle  p 
from  the  table  above  referred  to. 

Mars,  Venus,  and  Mercury. — These  planets  may  be  regarded  as 
spherical  in  the  computation  of  their  occupations,  and  we  shall, 
therefore,  have  to  consider  only  their  crescent  and  gibbous 
phases.  To  adapt  our  formula?  to  the  case  of  a  spherical  body, 
we  have  only  to  put  e  =  0,  or  c  —  1.  Since  in  this  case  we  are 
concerned  only  with  the  apparent  figure  of  a  partially  illuminated 
spherical  body,  we  may,  for  the  convenience  of  computation, 
assume  any  point  as  the  pole  of  the  planet ;  and  it  will  be  most 
natural  to  assume  the  point  which  is  the  pole  of  the  great  circle 
whose  plane  passes  through  the  sun,  the  earth,  and  the  planet. 
y  49  The  direction  of  this  pole  is  evidently  the 

same  as  that  of  the  line  joining  the  cusps 
of  the  partially  illuminated  disc.  This  makes 
ft  =  0,  B  —  0,  in  (615),  and,  consequently, 
V=  A  —  L  But,  as  the  adopted  equator  of 
the  planet  is  here  a  variable  plane,  we  can 
no  longer  use  the  form  (620)  for  finding  A. 
A  very  simple  and  direct  process  for  finding 
V  offers  itself.  Let  E,  S,  0,  Fig.  49,  repre- 


FORM    OF   A   PLANET  S   DISC. 


577 


sent  the  centres  of  the  earth,  the  sun,  and  the  planet;  S'O'O", 
the  great  circle  of  the  celestial  sphere  whose  plane  passes  through 
the  three  bodies ;  Sf  and  0',  the  geocentric  places  of  the  sun  and 
the  planet;  0",  the  heliocentric  place  of  the  planet.  Then  0'0" 
is  the  arc  heretofore  denoted  by  V,  and,  in  the  infinite  sphere,  is 
the  measure  of  the  angle  0'00"  =  SOE.  Putting  thenF=0'0", 
r  =  S'0',  and  also 


R'  =  SO  =  the  heliocentric  distance  of  the  planet, 
E  =  SE  =  «  "  «      earth, 


we  have 


=  — -  sin 


Fig.  50. 


We  might  find  V  directly  from  the  three  known  sides  of  the 
triangle  SOE;  but,  as  we  have  yet  to  find  p,  and  f  conies  out  at 
the  same  time  with  p  in  a  very  simple  manner,  it  will  be  prefer- 
able to  employ  the  above  form. 

To  find  p  and  7-,  let  S',  O ',  0",  Fig.  50,  be  the  three  places 
above  referred  to,  and  P  the  pole  of 
the  equinoctial.  Draw  O'Q  perpen- 
dicular to  the  great  circle  S'0'0". 
This  perpendicular  passes  through  the 
adopted  pole  of  the  planet,  and  we 
have  PO'Q=p,  or  PO'S'=9Q°  —  p, 
and  S'0'=  ?.  Hence,  denoting  by  df 
and  D  the  declination  of  the  planet 
and  the  sun,  and  by  a'  and  A  their 
right  ascensions  respectively,  the  spherical  triangle  PS'O'  gives 


cos  Y  =  sin  3'  sin  D  -f-  cos  d'  cos  D  cos  (a'  —  A) 
sin  Y  sin  p  =  cos  d'  sin  D  —  sin  3'  cos  D  cos  (a'  —  A) 
sin  Y  cos  p  =  cos  D  sin  (a'  —  A) 


(622) 


Hence,  introducing  an  auxiliary  to  facilitate  the  computation, 
both  p  and  V  will  be  found  by  the  following  formulae  : 


tan  F  =  tan  D  sec  (a'  —  A) 
tan  p  =  cot  (a'  —  A)  sin  (F  —  8')  sec  F 
R    sin  (a'  —  A)  cos  D 


_ 


cos^> 


(623) 


In  this  method  of  finding  V  we  do  not  determine  whether  it  is 
VOL.  L,  -37 


578 


OCCULTATIONS   OF   PLANETS. 


greater  or  less  than  90°.  This  is  of  no  importance  in  computing 
an  actual  observation,  but  only  in  predicting  the  phase  of  the 
planet,  whether  crescent  or  gibbous.  For  the  latter  purpose  we 
must  have  recourse  to  the  triangle  SEO  of  Fig.  49,  the  three 
sides  of  which  are  given  in  the  Ephemeris. 

The  value  of  V  being  found,  the  equation  (616)  will  be  used  to 
determine  the  apparent  outline  after  substituting  c  =  1  and 
w  =  90°,  whereby  it  becomes 

s2  =  u2  -f-  u2  sec2  V 

The  value  of  s  in  our  equations  is  supposed  to  be  given.  It 
will  be  most  convenient  to  deduce  it  from  the  apparent  semi- 
diameter  of  the  planet  when  at  a  distance  from  the  earth  equal 
to  the  earth's  mean  distance  from  the  sun,  which  is  the  unit 
employed  in  expressing  their  geocentric  distances  in  the  Ephe- 
meris. Thus,  denoting  the  mean  semidiameter  by  s0,  and  the 
geocentric  distance  by  r',  we  have  (Art.  128) 


s  = 


r' 


(624) 


and  SQ  may  be  taken  from  the  following  table : 


«o 

Authority. 

MERCURY  

3".  34 

LE  VERRIER,  Theory  of  Mercury. 

VENUS  ...         ...    

8  .55 

PEIRCE,  Am.  Ephemeris. 

IVlARS 

6  .05 

99  .70 

STRUVE,  Astr.  JVach.,  No.  139. 

81  .36 

BESSBL,  Astr.  Nach.,  No.  275, 

SATURN'S  RINGS     ..  .   .          

Outer  semi-major  axis  of  outer  ring 

Inner           "             "              "         " 
Outer          "            **           inner     " 
Inner          "            "             "         " 

187  .56 

165  .07 
161  .27 
124  .75 

«                «                 « 
{STRUVE,  Astr.  Nach.^o.  139, 
reduced  to  agree  with  BES- 
SEL'S  measures  of  the  outer 
diameter  of  the  outer  ring. 

353.  To  find  the  longitude  of  a  place  from  the  observed  contact  of 
the  moon's  limb  with  the  limb  of  a  planet. — In  the  following  investi- 
gation, it  is  assumed  that  the  quantities  p,  w,  V,  c,  are  known  for 
the  time  of  the  occupation.  They  may  be  computed  by  the 
above  methods  for  the  time  of  conjunction  of  the  moon  and 
planet,  and  regarded  as  constant  for  the  same  occultation  over 
the  earth  in  general. 


LONGITUDE. 


579 


Let  0,  Fig.  51,  be  the  apparent  centre  of  the  planet,  and  C 
the  point  of  contact  of  its  limb  with 
that  of  the  moon.  Let  OM  be  drawn 
from  0  towards  the  moon's  centre,  in- 
tersecting the  moon's  limb  in  D.  Since 
the  apparent  semidiameter  of  any  of 
the  planets  is  never  greater  than  31", 
it  is  evident  that  no  appreciable  error 
can  result  from  our  assuming  that  the 
small  portion  CD  of  the  moon's  limb 
coincides  sensibly  with  the  common 
tangent  to  the  two  bodies  drawn  at  C. 
If,  then,  the  planet  were  a  spherical 
body  with  the  radius  OD,  the  observed 
time  of  contact  would  not  be  changed.  We  may,  therefore, 
reduce  the  occultation  of  a  planet  to  the  general  case  of  eclipse 
of  one  spherical  body  by  another,  by  substituting  the  perpen- 
dicular OD  for  the  radius  of  the  disc  of  the  eclipsed  body.  Let 
s"  denote  this  perpendicular;  let  OA  and  OQ  be  the  axes  of  u 
and  v  respectively,  to  which  the  curve  of  illumination  is  referred 
by  the  equation  (616) ;  and  let  #  be  the  angle  QOD  which  the 
perpendicular  s"  makes  with  the  axis  of  v.  The  equation  of  the 
tangent  line  CD  referred  to  these  axes  is 


u  sin 
We  have  also  in  the  curve 


v  eos  &  = 


(625) 


dv 

—  —  —  tan 

du 


Differentiating  the  equation  (616),  therefore,  we  have 

(v  sin  M?\/  tan  #  sin  w\ 

u  cos  w  ------  -  1  1  cos  w  -f-  -  I 


I  u  si 


-}-  I  u  sin  w  -\- 


V  COS  W 


Htan  ft  cos  w  \ 
sm  w Jsec2F=  0 


By  means  of  this  equation,  together  with  (616)  and  (625),  we  can 
eliminate  u  and  u,  and  thus  obtain  the  relation  between  s  and  s". 
To  abbreviate,  put 


x  =  u  cos  w  — 
y  =  u  sin  w  -j- 


u  sin  10 

c 
v  cos  10 


530 
and  also 


OCCULTATIONS    OF    PLANETS. 


c'sintf' 


c'cos  ft'  =  cos  ft 


(626) 


then  the  three  equations  become 

x  cos  (ft'  —  10)  —  y  sin  (ft'  —  w)  sec2  V  =  0 


sn 


'  —  it?)  -j-  y  cos  (#'  —  w)  =  —  ? 


From  the  first  and  second  of  these  we  find 

s  sin  (ft'  —  w) 


y  = 


—  cos2  (//  —  10)  sin2  V~\ 
s  cos  (#'  —  10)  cos2  V 


—  cos2  (*'—  u?)  sin2  7] 
which  substituted  in  the  third  give 

5"  =  8cc'y\\  —  cos2  (#'  —  w)  sin2F] 
Hence,  if  we  put 
we  have 


sin  /  =  cos  (»>'  — 
5/f  =  s  .cc'  cos 


sinF 


(627) 


We  have  seen  (Art.  352)  that  in  all  practical  cases  we  may  take 
w  =  90°,  and,  therefore,  instead  of  (626)  and  (627)  we  may 
employ  the  following : 

tan  # 


tan  ft '  = 

c 

sin  /  =  sin  ft'  sin  F 
s  sin  ft  cos 


sin  ft' 


(628) 


If  the  occultation  of  a  cusp  of  Venus  or  Mercury  is  observed, 
we  have  at  once  str  =  s  cos  $  (for  the  axis  of  v  coincides  with 
the  line  joining  the  cusps),  and  we  do  not  require  V. 

The  value  of  s"  is  to  be  substituted  in  (486)  for  the  apparent 
semidiameter  of  the  eclipsed  body.  In  that  formula,  .//denotes 
the  apparent  semidiameter  at  the  distance  unity :  therefore,  we 
must  now  substitute  the  value 


sin  H  =  r'  sin  s" 


LONGITUDE.  581 

or,  by  (624)  and  (628), 

^       sin  5  sin  #  cos/ 

sm  IT  =  -  (629) 

sin*' 

Since  /  is  here  very  small,  we  may  put  tan  /=  sin  /,  and  the 
formula  for  L  (488)  becomes 

L  —  (2  —  C)  sin  /  ±  k 

«<I^C)«±j|^ft--Q'^S 
r#  r# 

Hence,  putting 

*  =  ft  +  (*-3*^p  (630) 

we  have 

L  =  (z  -  C)  ^?  ±  #  •          (631) 

r'g 

When  the  angle  $  is  known,  therefore,  the  preceding  formulae 
will  determine  L,  with  which  the  computation  will  be  carried 
out  in  precisely  the  same  form  as  in  the  case  of  a  solar  eclipse, 
Art.  329.  To  find  #,  let  OP,  Fig.  51,  be  drawn  in  the  direction 
of  the  pole  of  the  equinoctial;  then  we  have  POQ  =  p,  and, 
denoting  POM  by  §, 

#=Q-P 

and  Q  has  here  the  same  signification  as  in  the  general  equations 
(567),  as  shown  in  Art.  295 :  so  that  when  N  and  ^  have  been 
found  by  (568)  and  (569),  we  have  Q  =  N  -f  ^,  or 

#  =  N  +  4  —  p  (632) 

But  to  compute  $  by  (569)  we  must  know  Z/,  and  this  involves 
H,  which  depends  upon  $.  The  problem  can,  therefore,  be 
solved  only  by  successive  approximations;  but  this  is  a  very 
slight  objection  in  the  present  case,  since  the  only  formulae  to  be 
repeated  are  those  for  L  and  ^,  and  the  second  approximation 
will  mostly  be  final.  It  can  only  be  in  a  case  such  as  the  occul- 
tation  of  Saturn's  ring,  where  the  outline  of  the  eclipsed  body  is 
very  elliptical,  and  especially  when  the  contact  occurs  near  the 
northern  or  southern  limb  of  the  moon,  that  it  maybe  necessary 
(for  extreme  accuracy)  to  compute  H  &  second  time  and,  conse- 
quently, ^  a  third  time. 

The  formula  (629)  is  adapted  to  the  general  case  of  an  ellip- 


OCCULTATIONS    OF    PLANETS. 


soidal  body  partially  illuminated,  the  point  of  contact  being  on 
the  defective  limb.  When  the  point  of  contact  is  on  the  full 
limb,  we  have  only  to  putF=  0,  and  the  formula  becomes 


sn 


(633) 


and  for  the  full  limb  of  a  spherical  planet  (Venus,  Mercury,  and 
Mars)  we  have  H  =  SQ. 

In  the  first  approximation  we  may  take  L  =  ±  k. 

354.  Sometimes  it  may  not  be  known  from  the  record  of  the 
observation  whether  the  point  of  contact  is  on  the  full  or  the 
defective  limb  of  the  planet.  This  might  be  determined  by  the 
method  of  Art.  350  ;  but,  since  that  method  supposes  the  position 
angle  pf  to  be  given,  which  we  do  not  here  employ,  the  following 
more  direct  and  simple  process  may  be  used.  In  that  article  the 
common  point  of  tangency  of  the  two  curves  of  the  full  and 
defective  limbs  was  determined  by  the  condition 

cos  w 

ul  sin  w  -f-  vl  -  =  0 

in  which  u^  and  i\  denote^  the  co-ordinates  of  the  point  of  tan- 
gency.. In  the  notation  bf  Art.  353  this  is  simply  y^  =  0  ;  and 

since  we  have 

s  cos  (?9j  —  w)  cos2  V 

y^~  v/[l  _  cos2  (7^—  w)  sin2  F] 
it  follows  that  we  must  have 

cos  (tfj  —  w)  =  0  or  ^  —  w  T  90° 

Hence,  when,  as  in  our  present  application,  we  take  w  =  90°,  we 

have 

^  =0  or  ^  =  180° 

Hence  a  point  is  to  be  regarded  as  on  the  east  limb  for  values  of  & 
between  0°  and  180°,  and  on  the  west  limb  for  values  of  d-  between  180° 
and  360°  ;  and  (Art.  350)  the  east  or  the  west  limb  is  defective  accord- 
ing as  Vis  between  0°  and  180°  or  between  180°  and  360°. 

But,  since  sin  #'  and  sin  &  have  the  same  sign,  we  deduce  from 
this  a  still  more  simple  rule  ;  for  we  have  sin  £  —  sin  &'  sin  V, 
whence  it  follows  that  the  observed  point  is  on  the  defective  limb 
when  sin  %  is  positive,  and  on  the  full  limb  when  sin  %  is  negative. 


LONGITUDE.  583, 

355.  In  the  cases  of  the  planets  Neptune,  Uranus,  and  the 
asteroids,  the  occultation  of  their  centres  will  be  observed,  and 
it  will  be  most  convenient  to  compute  by  the  method  for  a  fixed 
star,  only  substituting  for  n  the  difference  of  the  moon's  and 
planet's  horizontal  parallaxes  —  that  is,  the  relative  parallax  —  in 
the  formulae  for  x  and  ?/,  Art.  341. 

This  artifice  of  using  the  relative  parallax  may  also  be  used 
with  advantage  for  Jupiter  and  Saturn. 

Having  thus  found  x  and  y  as  for  a  fixed  star,  we  shall  have, 
in  the  preceding  method, 


(634) 


the  other  formulae  remaining  unchanged. 


EXAMPLE  1.  —  Several  occultations  of  Saturn's  Ring  were  ob- 
served by  Dr.  KANE  at  Van  Rensselaer  Harbor  on  the  northwest 
coast  of  Greenland  during  the  second  Grinnell  Expedition  in 
search  of  Sir  JOHN  FRANKLIN.*  The  first  of  these  was  as 
follows  : 

1853  December  12th,  Van  Rensselaer  Mean  Time 
Immersion,  contact  of  last  point  of  ring,    .     .     .     14*  20"*  48*.8 
Emersion,        «  "        "  "...     14   54    18.3 

The  assumed  longitude  of  the  place  of  observation  was  w=4A43m  32* 
west  of  Greenwich.     The  latitude  was  <p  =  78°  37'  4",  whence 

Jog  p  sin  <?'  =  9.989862  log  p  cos  <p'  =  9.296642 

I.  From  the  Nautical  Almanac  we  take  for  1853  Dec.  12,  19*, 
p  =  —  2°  37'.3         I  —  24°  0'.4         whence  log  c  =  log  sin  I  =  9.6094 
and  from  page  578,  the  outer  ring  only  being  observed, 
S0  =  187"  .56  log  sin  SQ  =  6.9587 


*  ''Astronomical  Observations  in  the  Arctic  Seas  by  ELISHA  KENT  KANE,  M.D.. 
U.S.N.  Reduced  and  discussed  by  CHARLES  A.  SCHOTT,  Assistant  U.S.  Coast 
Survey."  Published  by  the  Smithsonian  Institution,  May,  1860. 


584 


OCCULTATIONS    OF    PLANETS. 


II.  We  shall  compute  the  elements  of  the  occultation  for  the 
centre  of  the  planet  for  the  Greenwich  hours  18*,  19*,  and  20\ 
For  these  times  we  take  the  following  quantities  from  the 
Nautical  Almanac,  applying  to  them  the  corrections  determined 
by  Mr.  SCHOTT  from  the  Greenwich  observations  of  this  date : 

Moon. 


Gr.  T. 

a 

j 

7T 

18* 

3*  36-  55'.23 

-f  18°  2'  47".5 

54'  7".68 

19 

38  53.92 

12  13  .9 

7  .22 

20 

40  52.81 

21  35  .7 

6  .76 

Saturn. 


a' 

6' 

TT' 

logr' 

18* 

.  3*  39"1  9'.88 

+  17°  14'  28".4 

1".05 

0.9126 

19 

9.16 

26  .5 

20 

8.44 

24  .5 

The  corrections  applied  to  the  Nautical  Almanac  values  to 
obtain  the  above  are  AOC  =  —  0*.22,  A£  =  —  5".0,  *a' =  -f  OM5, 
A<J'=  —  8". 9,  AT  = -f- 0''.3,  this  last  correction  being  derived 
from  Mr.  ADAMS'S  Table  in  the  Nautical  Almanac  for  1856. 

We  shall  use  the  relative  parallax,  and  compute  as  for  a  fixed 
star,  taking  n  —  K'  for  TT,  namely 


7T 

18* 

54'  6".73 

19 

6  .17 

20 

5  .71 

whence  we  find  for  the  moon's  co-ordinates, 


Gr.  T. 

X 

x' 

y 

y' 

18* 

-0.59152 

-f  0.52457 

+  0.89382 

-fO.17436 

19 

—  0.06690 

+  0.52466 

-f-  1.06817 

-f  0.17434 

20 

-f  0.45781 

+  0.52475 

+  1.24250 

+  0.17432 

LONGITUDE. 


585 


and,  taking  z  =  r  = 


1 


sin 


for  19*,  as  sufficiently  accurate, 


2  =  63.54 
III.  For  the  co-ordinates  of  the  place  of  observation : 


Local  mean  time  t 

t  -f-  w 

Local  sid.  time  /x 

Immersion. 

Emersion. 

14*  20-  48'.8 
19     4    20.8 
117°   4'  59".7 

14*  54"  18'.3 
19   37    50.3 
125°  28'  44".7 

and  hence,  by 

the  formulae  on  p.  550, 

ff 
y 

C 
z  —  C 

-f    0.17529 
-f    0.90575 
+    0.38 
63.16 

+    0.18685 
+    0.91363 
+    0.35 
63.19 

IV.  Assuming  now  two  epochs  corresponding  nearly  to  the 
times  of  observation,  the  remainder  of  the  computation  in  extenso 
is  as  follows : 


Assumed  T0  j 

19*.07  == 
19*  4*  12' 

19*.63  = 
19*  37™  48*. 

XQ 

-  0.03017 

+  0.26365 

y0 

+  1.08037 

-j-  1.17800 

x0  —  £  =  m  sin  M 

-  0.20546 

-f-  0.07680 

yQ  —  y  =  m  cos  M 

+  0.17462 

-f  0.26437 

M 

310°  21'  38" 

16°  11'  56" 

log  m 

9.43079 

9.43980 

x*  =  n  sin  JV 

-f  0.52467 

-f  0.52472 

y'  =  n  cos  JV 

-j-  0.17434 

-f  0.17433 

N. 

71°  37'  10" 

71°  37'  20" 

logn 

9.74263 

9.74266 

Then,  for  a  first  appproximation,  by  the  formula 

sin  &  —  —  

and  observing  that  the  immersion  is  here  an  interior  contact  and 
the  emersion  an  exterior  contact,  we  have 


586 


OCCULTATIONS    OF    PLANETS. 


Immersion. 

Emersion. 

log  sin  (M  —  N) 

n9.93188 

W9.91559 

log  m 

9.43079 

9.43980 

(L  =  q:  k)         ar.  co.  log  L 

wO.56441 

0.56441 

log  sin  4 

9.92708 

n9.91980 

** 

57°  43'.2 

303°  45'.5 

N-p 

74    14.5 

74    16.6 

N  -f  *  —  P  =  '* 

131    57.7 

18      0.1 

log  tan  # 

nO.0462 

9.5118 

logc 

9.6094 

9.6094 

log  tan  #' 

nO.4368 

9.9024 

log  sin  # 

W9.8713 

9.4900 

ar.  co.  log  sin  #' 

nO.0273 

0.2047 

(•  >  •  r  v 
1    C  1  11       O 

7.8465 

7.8467 

r'       1  81  l  So 

flog* 

7.7451 

7.5414 

a 

0.00556 

0.00348 

=P  A- 

-  0.27264 

-f-  0.27264 

a  qr  k  =  L 

—  0.26708 

+  0.27612 

log  L 

W9.42664 

9.44110 

Applying  the  difference  between  log  .L  and  log  k  to  log  sin  ^,  we 

find,  for  our  second  approximation, 

Corrected  log  sin  4 

9.93603 

9.91429 

4 

59°  39'.6 

304°  49'.5 

"                       # 

133    54.1 

19      4.1 

log  tan  ^ 

nO.0167 

9.5387 

log  tan  #' 

wO.4073 

9.9293 

log  sin  ?9 

w9.8577 

9.5141 

ar.  co.  log  sin  #' 

wO.0310 

^1887 

7.8465 

7'.8467 

Corrected  log  a 

7.7352 

7.5495 

"                a 

.  0.00543 

0.00354 

X 

-  0.26721 

+  0.27618 

log  i 

n9.42685 

9.44119 

Final  value  of  log  sin  4 

9.93582 

n9.91420 

log  cos  4- 

9.70403 

9.75688 

*  The  angle  4,  is  to  be  taken  so  that  L  cos  4  shall  be  negative  for  immersion  and 

positive  for  emersion,  Art.  829. 

t           ,  sin  //       sin  •&   z  —  C 
f  Putting  a  —  (z  —  f\             —                      s  .  s~ 

sintf'       r' 


LONGITUDE. 


587 


Immersion. 

Emersion. 

_AJ,cos4 

7i2.94455 

3.01171 

,           t>                   to             w 

Jim  cos  (  Jf  —  JV) 

Incr  />  —  Iftjr 

7i2.95956 

3.00741 

lOg  G           lOg                       ^ 

5 

—  880M 

-f  1027-.3 

C 

-  911  .1 

-f  1017  .2 

b  —  c  =  r 

+    31.0 

-f-      10.1 

Gr.  Time  of  obs.  =  T0  -f  r  =  21 

19*    4-43-.0 

19*  37-  58-.1 

T  f  rr=  W 

4  43    54.2 

4   43    39.8 

If  now  we  wish  to  form  the  equations  of  condition  for  deter- 
mining the  effect  of  errors  in  the  data,  we  proceed  precisely  as 
in  the  case  of  a  solar  eclipse,  page  533,  and  find 


log  v  tan  4 
log  v  sec  4, 

Immersion. 

Emersion. 

0.5341 

0.5983 

wO.4596 
0.5454 

where  log  v  =  log  -    -  =  0.3023.     Hence,  neglecting  the  terms 

depending  on  the  correction  of  the  parallax  and  of  the  eccen- 
tricity of  the  meridian,  the  equations  of  condition  are 

(Im.)     oij  =  4*  43-  54'.2  —  2.001  r  -f  3.421  0  —  3.965  r  A* 
(Em.)    <w,  =  4   43    39  .8  —  2.001  r  —  2.881  0  -f  3.511  r^k 

Eliminating  •&  from  these  equations,  we  have 

<»,  =  4*  43-  468.4  —  2.001  r  +  0.092  TT  A/; 

An  error  of  V  in  the  moon's  semidiameter  (represented  by  ;TA£) 
would,  therefore,  have  no  sensible  effect  upon  this  combined 
result ;  and  since  ?  must  also  be  very  small,  as  we  have  corrected 
the  places  of  the  moon  and  planet  by  the  Greenwich  observations, 
we  can  adopt,  as  the  definite  result  from  this  observation, 

^  =  4*  43-  46'.4 

It  will  be  observed  that  in  this  example  OUDEMANS'S  value, 
k  =  0.27264,  has  been  employed ;  but  our  final  equation  shows 
that  the  result  would  have  been  sensibly  the  same  if  we  had 
taken  the  usual  value  0.27227  ;  for  the  reduction  of  the  result  to 
that  which  the  latter  value  of  k  would  have  given  is  only 
0.092  X  3247  X  (-  0.00037)  =  —  OM1. 


588 


OCCULTATIONS    OF    PLANETS. 


EXAMPLE  2. — The  occultation  of  Venus,  April  24,  1860,  was 
observed  at  the  U.  S.  Military  Academy,  West  Point  (o>=4A55m  51*, 
<p  =  41°  23'  31".2),  and  at  Albany  (w  =  4*  54™  59S.4,  <p  =  42°  39' 
49".5),  as  follows : 

Immersion. 

First  contact,  planet's  full  limb 

Disappearance  of  cusp 

The  observations  were  made  with  the  large  refractors  of  the 
West  Point  and  Dudley  observatories. 

I.  To  find  p  for  the  cusp  observations,  we  have  for  the  Green- 
wich time  13/1.478,  which  is  the  mean  of  the  times  of  the  obser- 
vations at  the  two  places,  and  will  serve  for  both, 


West  Point. 
Sid.  time. 

Albany. 
Mean  time. 

10*  46-  53«.35 
10   47    47.80 

8*31™    1*.9 
8   31    54.2 

Planet,  a'  =  78°  38'.6 
Sun,       ^1=32    45.5 


d'  =  25°  59M 
D  =  13    12  .9 


whence,  by  (623), 
and,  from  p.  578, 


p  =  _  7 


log  lin  SQ  —  5.6175 


II.  We  shall  compute  the  moon's  co-ordinates  only  for  the 
Greenwich  times  13A.4  and  13*. 5.  For  these  times  the  American 
tykemeris  furnishes  the  following  data : 


Moon. 


Gr.  T. 

a 

6 

IT 

13*.4 
13  .5 

79°  12'  16".8 
79  15  58  .5 

-f  26°  43'  1".6 
26  43  4  .3 

*  57'  6".6 

57  6  .7 

Venus. 


a' 

ff 

log  r' 

13*.4 
13  5 

78°  38'  23".3 

78    38  40  .7  * 

+  25°  59'  2".5 
25    59  4  .3 

9.9193 
9.9193 

Hence,  by  the  formulae  of  I.  and  II.,  p.  452,  we  find 


a 

LONGITUDE. 
d 

iog^ 

13\4 
13.5 

78°  38'  17".2 
78    38  34  .0 

4-  25°  58'  54".5 
25    58  56  .3 

9.9987 
9.9987 

589 


X 

X> 

y 

y1 

13.4 
13.5 

4-  0.531695 
4-  0.585085 

2  = 

4-  0.53390 
=  60.19 

4-  0.773681 
4-  0.774161 

4-  0.00480 

III.  For  the  co-ordinates  of  the  places  of  observation 


Local  mean  time  t 

*  + 

log  p  sin 


West  Point. 

Albany. 

Full  limb. 

Cusp. 

Full  limb. 

Cusp. 

me  t 

8*  33™  43'.  72 

8*  34™  38'.  02 

8*  31™  K90 

8*  31™  54'.  20 

+  '« 

13   29    34.72 

13  30     29.02 

13   26    1  .30 

13   26    53.60 

/" 

161°  43'   20".3 

161°  56'  57".0 

161°   2'  44".3 

161°  15'  50".9 

in$' 

9.818064 

9.828792 

OS  (j)' 

9.875814 

9.867157 

£ 

4-  0.745828 

4-  0.746178 

4-  0.730013 

4-  0.730378 

n 

4-  0.551616 

4-  0.552909 

4-  0.563428 

4-  0.564641 

c 

-f  0.37 

4-  0.37 

4-  0.38 

4-  0.38 

—  c 

59.82 

59.82 

59.81 

59.81 

IV.  Assuming  T0=  13*.45,  we  find,  for  this  time, 


ar0 

4-  0.558390 

y<> 

4-  0.773921 

TO  sin  M 

-0.187438 

—  0.187788 

—  0.171623 

^-0.171988 

m  cos  M 

4-  0.222305 

+  0.221012 

4-  0.210493 

4-  0.209280 

M 

319°  51'  50" 

319°  38'  47" 

320°  48'  30" 

320°  35'  11" 

log  m 

9.463563 

9.462425 

9.433915 

9.432783 

N 

89°  29'  6" 

log  n 

9.727480 

Then,  for  the  observations  of  the  full  limb,  we  have  for  both 
places,  by  (631),  putting  H  —  s0, 


log  (z  -  C) 

ar.  co.  log  r'ff 

k  —  0.27264  constant 

0.00082 log 

k'  =0.27346 

0.00299 

L=  0.27645 


1.7768 
0.0820 
5.0542 
6.9130 


log  sin 


log(l) 


1.7768 
0.0820 
5.6175 


7.4763 


590 


OCCULTATIONS    OF   PLANETS. 


West  Point. 

Albany. 

M—N 

230°  22'  44" 

231°  19'  24" 

4 

234      6    57 

230      4    55 

T 

-f     2-  37'.7 

52'.7 

T0 

13*  27m    0'. 

13*  27-  C*. 

T 

13   29    37.7 

13   26    7.3 

T—t  =  w 

4   55    54.0 

4   55    5.4 

For  the 

observations  of  the  cusps  we  can  employ  the  preceding 

values  of 

\j/  as  a  first  approximation  ;  and  hence  we  proceed  as 

follows  : 

West  Point, 

Albany. 

N 

+  4-^Hdfa* 

331°  3'.4 

327°  1'.3 

log  cos  ft 

9.9421 

9.9237 

log(l) 

7.4763 

7.4763 

7.4184 

7.4000 

* 

0.00262 

0.00251 

H 

0.27346 

0.27346 

L 

0.27084 

0.27095 

M—N 

230°  9'  41" 

231°  6'  5" 

log 

sin  (M  —  N) 

n9.885278 

7i9.891124 

log  m 

9.462425 

9.432783 

ar.  co.  log  L 

0.567287 

0.567111 

log  sin  4 

w9.914990 

n9.891018 

4 

235°  18'.5 

231°  5'.0 

Corrected  # 

332    14.9 

328    1.4 

log  cos  »9 

9.9469 

9.9285 

log  (1) 

7.4763 

7.4763 

7.4232 

7.4048    * 

0.00265 

0.00254 

Corrected  £ 

0.27081 

0.27092 

ar.  co.  log  L 

0.567335 

0.567159 

Corrected  log  sin  4, 

n9.915038 

9.891066 

r 

+      3-  33«.7 

0*.4 

TQ  +  r=T 

13*  30W  33*.7 

13*  26-  59«.6 

T  —  t  —  t» 

4   55    55.7 

4   55      5.4 

Finally,  if  we  wish  to  form  the  equations  of  condition  for 
correcting  these  results  for  errors  in  the  data,,  including  an  error 
in  the  planet's  semidiameter,  we  proceed  as  for  an  eclipse  of  the 


TRANSITS    OF    VENUS    AND    MERCURY.  591 

sun,  p.  533.  For  the  full  limb  we  have  only  to  substitute  A50  for 
&H;  but  for  the  cusp  we  must  evidently  substitute  ASO  cos  $  for 
A  H.  It  will  be  more  accurate  to  restore  r'g  in  the  place  of  r', 
since  g  here  differs  sensibly  from  unity.  We  shall  thus  find 


w'=  4*55™  54-.0  —  1.967  r  +  2.720  #  —  3.358  rAA  —  4.061  ASO 
w'=  4  55  55  .7  —  1.967  r  +  2.844  0  —  3.459  TTAA  4-  3.697  ASO 
«>"=  4  55  5  .4  —  1.967  r  +  2.352  fl  —  3.067  *  A  A-  —  3.704  ASO 
«"=  4  55  5  .4  —  1.967  r  +  2.438  #  —  3.134  TT  AA  -f  3.349  ASO 

where  to'  and  a/'  denote  the  true  longitudes.     Hence,  also, 


w'  —  a>"  =  4-  48'.6  4-  0.368  *  —  0.291  TT  A*  —  0.357  ASO 
«/  —  «/'  =  -f-  50  .3  +  0.406  #  —  0.325  TT  AA  +  0.348  ASO 

and  the  mean  is 

«/  _  «,"'=  -f-  49'.5  4-  0.387  *  —  0.308  -  AA  —  0.005  ASO 

The  effect  of  an  error  in  s0  upon  the  difference  of  longitude  of 
the  two  places  is,  therefore,  insensible  ;  but,  to  eliminate  &  and 
;TA&,  observations  of  the  emersion  should  also  be  used.  The 
effect  of  f  and  ft  upon  a)f  and  to"  can  only  be  eliminated  by 
means  of  observations  of  the  moon's  place  at  a  standard  observa- 
tory on  the  day  of  the  observation,  as  we  have  already  shown  in 
other  examples. 

TRANSITS  OF  VENUS  AND  MERCURY. 

356.  The  transits  of  Venus  and  Mercury  may  be  computed  by 
the  method  for  solar  eclipses,  substituting  the  planet  for  the 
moon.  In  the  formulae  (486),  (487),  &c.,  we  must  employ 

for  Venus,      k  =  0.9975 
for  Mercury,  A  =  0.3897 

which  are  the  values  which  result  from  the  apparent  semi- 
diameters  of  these  planets  adopted  on  p.  578. 

Since  b  is  no  longer  a  small  quantity,  it  will  be  necessary  to 
employ  the  exact  formulae  (479)  instead  of  (481). 

The  longitude  of  a  place  at  which  the  transit  is  observed  may 
be  computed  from  each  of  the  four  contacts  of  the  limb  of  the 
sun  and  planet,  by  the  formulae  of  Art.  329.  These  observations, 
however,  are  of  little  use  in  determining  an  unknown  longitude, 
on  account  of  the  great  effect  of  small  errors  in  the  assumed 


592  TRANSITS  OF  VENUS  AND  MERCURY. 

parallax  upon  the  computed  time  ;  but,  on  the  other  hand,  when 
the  longitude  is  previously  known,  each  observation  furnishes 
an  equation  of  condition  of  the  form  (584)  for  determining  the 
correction  of  the  parallax.  In  developing  this  equation,  however, 
we  supposed  g  =  1,  in  the  formula  (486),  and  we  must,  therefore, 
here  restore  the  true  value.  "We  may  take 


in  which  it  and  TT'  are  the  assumed  horizontal  parallaxes  of  the 
planet  and  sun  respectively  at  the  time  of  the  observation. 
Instead  of  the  form  for  I  employed  on  p.  449,  we  shall  now  take 
the  more  correct  form 

i=  *L±k 

r'gx       g 
If  we  denote  the  sun's  semidiameter  at  the  time  of  the  obser- 

TT" 

vation  by  s',  that  of  the  planet  by  s,  we  have  s'  =  —  ,  5  =  ?:&, 
and  hence 

7  —  *'—  s 


and  instead  of  (581)  we  shall  have 

7         A(s'=h  s)         s'  ±  S    ATT 
AX/  =  At  =  —  ----  -  • 

git  gn         TT 

Omitting  the  term  depending  upon  A?e,  which  can  never  be 
appreciable  in  the  transits  of  the  planets,  the  equation  (582)  will 
now  become 

w'  —  at  =  —  vf  -f-  v  tan  4  .  &  -f-  '         -  A(V  ±  s) 

(f  _(_  to  —  TJ  —  *  tan  *  -  -  sec  4!  ATT        (635) 

where  f  and  &  have  the  signification  (583)  ;  to'  is  the  true  longi- 
tude, and  a)  that  which  is  computed  from  the  observation. 

Since,  by  KEPLER'S  laws,  the  ratio  of  the  mean  distances  of  any 

two  planets  is  accurately  known  from  their  periods,  the  ratio  - 

^0 

is  also  known,  and  will  not  be  changed  by  substituting  the  cor- 
rected values  it  +  AT:  and  TTO  -f  ATTO  :  in  other  words  we  shall  have 

A:r          ic  xn 

—  •  =  —  or 


0 


TRANSITS    OF    VENUS    AND    MERCURY.  593 

The  discussion  of  all  the  equations  of  condition  of  the  form 
(635)  will,  therefore,  give  not  only  the  correction  AT:  of  the 
planet's  parallax,  but  also,  by  the  last-mentioned  relation,  that 
of  the  solar  parallax.* 

The  transits  of  Venus  will  afford  a  far  more  accurate  deter- 
mination of  this  parallax  than  those  of  Mercury ;  for,  on  account 
of  its  greater  proximity  to  the  earth,  the  difference  in  the  dura- 
tion of  the  transit  at  different  places  will  be  much  greater,  and 
the  coefficient  of  A;T  in  the  final  equations  proportionally  great. 

Although  the  general  method  for  eclipses  may  also  be  ex- 
tended to  the  prediction  of  the  transits  of  the  planets  (by  Art. 
322),  yet  it  is  more  convenient  in  practice  to  follow  a  special 
method  in  which  advantage  is  taken  of  the  circumstance  that 
the  parallaxes  of  both  bodies  are  so  small  that  their  squares  and 
higher  powers  may  be  neglected.  LAGRANGE'S  method  for  this 
purpose  is  the  most  simple,  and,  in  the  improved  form  which  I 
shall  give  to  it  in  the  following  article,  most  accurate. 

357.  To  predict  the  times  of  ingress  and  egress  for  a  given  place. — - 
We  first  find  the  times  of  ingress  and  egress  for  the  centre  of  the 
earth,  from  which  the  times  for  any  place  on  the  surface  are 
readily  deduced. 

Let  a,  d,  a',  d'  be  the  right  ascensions  and  declinations  of  the 
planet  and  the  sun  for  an  assumed  time  770, 
at  the  first  meridian,  near  the  time  of  con- 
junction.    Let  m  denote  the  apparent  dis- 
tance of  the  centres  at  this  time.     Let  Sf 
and  S,    Fig.  52,  be  the   geocentric   places 
of  the  centres  of  the  sun  and  planet,  P  the   s, 
pole ;  then,  denoting  the  angles  PS'S  and 
PSS'  by  P'  and  180°  -  P,  the*  triangle  PSS' 
gives 

sin  J  m  sin  J  (P  -f-  P')  =  sin  $  (a  —  a')  cos  £  (3  -j-  3') 
sin  i  m  cos  £  (P  -f-  P')  =  cos  J  (a  —  a')  sin  £  (3  —  3') 

But,  since  \m  is  at  the  time  of  a  contact  only  about  8',  we 
may  without  appreciable  error  substitute  it  for  its  sine,  and, 

*  Another  method  of  forming  the  equations,  apparently  shorter,   but  in  reality, 
where  many  observations  are  to  be  reduced,  not  more  convenient  than  the  rigorous 
method,  will  be  found  in  ENCKE'S  Die  Entfernung  der  Sonne  von  dcr  Erdc,  aus  dem 
Venusdurchgange  von  1761  hergeleitet;  and  Der  Venusdurchgang  von  1769. 
VOL.  I.— 38 


594  TRANSITS  OF  VENUS  AND  MERCURY. 

writing  M  for  J-  (P  -f-  P'),  we  may  regard  the  following  equations 
as  practically  exact : 

msin  M  =(a  —  a')  cos<50  |    ^686j 

mcosM  =    <J —  d'  f 

in  which  30  =  J  ((5  -f  <^'). 

Now,  let  the  required  time  of  contact  be  T  =  T0  +  r,  and  put 

«  =  the  relative  hourly  motion  of  the  two  bodies  in  right 

ascension, 

=  the  planet's  hourly  motion  —  the  sun's, 
d  —  the  relative  hourly  motion  in  declination, 

then  at  the  time  T  the  differences  of  right  ascension  and  de- 
clination are  a  —  a'  -f  ar  and  d  —  3'  +  dr.  If  further  we  put 

s,  sf  =  the  apparent  semidiameters  of  the  planet  and  sun, 
respectively, 

the  apparent  distance  of  the  centres  at  the  time  T  is  sf  ±  5,  the 
lower  sign  being  employed  for  inner  contacts ;  and  if  the  value 
of  M  at  this  time  is  §?  we  have 

(sr  ±  s~)  sin  Q  =  (a  —  a')  cos  d0  -j-  a  cos  <50 .  v 
(s'  ±  s)  cos  Q  =  d  —  <5'  -f-  dr 


Putting,  therefore, 

n  sin  N  =  a  cos  <50 
tt  cos  N  =  d 
we  have 

(Y  dz  s)  sin  §  =  m  sin  Jf  -f  w  sin  N.  r 
(sr  ±  5)  cos  Q  =  m  cos  M  -\-  n  cos  JV .  T 

which,  solved  in  the  usual  manner,  give 

m  sin  (M  —  JV) 

sm      =  —    — - 


:  _  _        _TN 

T  ^^ Cos  4, cos  (Jf  —  N) 

n  n         ^ 


}    (637) 


(638) 


where  cos  ^  is  to  be  taken  with  the  negative  sign  for  ingress 
and  with  the  positive  sign  for  egress.  The  angle  Q  is  (as  in 
eclipses)  the  position  angle  of  the  point  of  contact. 


TRANSITS    OF    VENUS    AND    MERCURY.  595 

The  formula  (636),  (637),  and  (638)  serve  for  the  complete 
prediction  for  the  centre  of  the  earth. 

To  find  the  time  of  a  contact  for  any  point  of  the  surface  of 
the  earth,  let  m  be  the  geocentric  apparent  distance  of  the 
centres  of  the  two  bodies  at  any  given  time ;  m'  the  apparent 
distance,  at  the  same  time,  as  seen  from  a  point  on  the  earth's 
surface  in  latitude  <p  and  longitude  M  ;  it  and  /r'  the  equatorial 
horizontal  parallaxes  of  the  planet  and  sun  respectively ;  £  and  £' 
their  geocentric  zenith  distances ;  />  the  radius  of  the  earth  for 
the  latitude  (p.  The  apparent  zenith  distances  are  £  -f-  pit  sin  £ 
and  £'  -f-  p  n'  sin  £  :  these  approximations  being  quite  exact 
where  the  parallaxes  are  so  small.  Let  Z,  Fig.  52,  be  the 
geocentric  zenith  of  the  place,  S  and  Sf  the  true  places  of  the 
bodies.  The  distance  SS'  =  m  will  become  the  apparent  dis- 
tance m'  if  we  increase  the  sides  ZS  and  ZSf  by  f>7ism£  and 
jOTr'sin^';  and,  if  we  regard  these  small  increments  as  differen- 
tials, we  shall  have,  by  the  first  equation  of  (46), 

m'  —  m  =  —  /OTT  sin  £  cos  S  -f-  px'  sin  £'  cos  Sf 

where  S  =  180°  —  ZSS',  and  Sf  =  ZS'S. 

Let  S0  be  the  middle  point  of  the  arc  SS',  and  denote  the 
angle  ZS0S  by  S0,  the  arc  ZS0  by  £0 ;  then  we  have 

—  sin  £  cos  S  =  sin  £  m  cos  £0  —  cos  £  m  sin  £0  cos  S0 
sin  £'cos  8'  =  sin  £  m  cos  £„  -|-  cos  J  r>i  sin  £0  cos  $0 

which  give 

m'  —  m  =  p  \_(K  -f-  TT')  sin  £  m  cos  £0  —  (n  —  TT')  cos  £  m  sin  C0  cos  $0] 
If  then  g  and  7-  are  determined  by  the  conditions 

}    (639) 
m'  —  m  =  gp  (sin  f  cos  £0  —  cos  f  sin  C0  cos  SQ) 


g  sin  y  =  (;r  -}-  TT')  sin  $  m 
</  COS  p  =  (TT  —  ?r')  COS  £  m 

we  have 


Produce  the  arc  S'S,  and  take  S0G  =  90°  +  r.    Then,  denoting 
the  arc  ZG-  by  ^,  the  triangle  ZGS0  gives 

cos  A  =  —  sin  f  cos  C0  -f-  cos  r  8in  C0  cos  ^ 
and  the  expression  for  mf  becomes 

mf  =  m  —  gp  cos  A  (640) 


596  TRANSITS    OF    VENUS    AND    MERCURY. 

This  remarkably  simple  form  was  iirst  given  by  LAGRANGE,* 
with  the  difference  only  that  he  regarded  the  earth  as  a  sphere, 
which  amounts  to  supposing  o  to  be  constant.  Under  this  sup- 
position, it  follows  from  the  equation  that,  at  any  given  time,  the 
apparent  distance  of  the  bodies  is  the  came  for  all  places  on  the  surface 
of  the  earth  which  have  the  same  value  of  / ;  that  is,  for  all  places 
whose  zeniths  are  in  a  small  circle  described  from  the  point  G  as  a  pole 
with  the  polar  distance  ZG  =  L 

The  computation  of  m'  will,  therefore,  be  extremely  simple 
after  the  position  of  the  point  G  is  determined.  The  quantity  Y 
is  determined  by  (639),  for  which,  however,  we  can  take 

tan  Y  = -7  tan  £  m 

V   (641) 

Let  A  and  D  denote  the  right  ascension  and  declination  of  the 
point  G.  Those  of  the  point  S0  are  very  nearly  a0=  J(a  -f  a') 
and  ^0=  J(o  -f  d'):  so  that  in  the  triangle  P8QGr  we  have  the 
angle  SQPG==A~  <x0,  the  side  PS0=  90°  —  <J0,  and  for  the  angle  - 
PS0G  we  can  take  M=±(PSG  +  PS'G)  as  in  (636).  Hence 
we  have 

cos  D  sin  (A  —  o0)  =       cos  y  sin  M  "| 

cos  D  cos  (A  —  a0)  =  —  cos  <\  sin  Y  —  sin  <50  cos  Y  cos  M    V    (642) 

sin  D  =  —  sin  <50sin  Y  +  cos  $0  cos  y  cos  M  ) 
or,  adapted  for  logarithms, 

/sin  7^=  sin  ^  cos  D  sin  (A — a())  =       cos^sinJf     "j 

/cos  F=  cos  Y  cos  M      cos  D  cos  (A  —  a0)  =  —  /sin  («50  -f-  F}  >  (642*) 

smD=      /cos(o0-f-.F)j 

For  any  given  time  T7,  therefore,  we  can  find  m  and  M  by 
(636),  then  Y  and  g  by  (641),  and  hence  the  values  o€  A  and  D  by 
(642).  Now,  let  ii  be  the  sidereal  time  (at  the  first  meridian) 
corresponding  to  T7,  and  put 

0  =  ft  —  A 

then,  in  the  triangle  PGZ,  we  have  the  angle  GPZ=  G  —  o>, 
and  hence,  <pf  being  the  geocentric  latitude  of  Z, 

cos  A  =  sin  <f>'  sin  D  -j-  cos  <p'  cos  D  cos  (0  —  a/)  (643) 

with  which  the  value  of  m!  will  be  found  by  (640). 

*  Memoirs  of  the  Berlin  Academy  for  1766.  The  above  extremely  simple  demonstra- 
tion I  suppose  to  be  new. 


TRANSITS  OF  VENUS  AND  MERCURY.  597 

Iii  order  to  apply  these  formulae  in  predicting  the  time  of  a 
contact  at  a  given  place,  we  observe,  first,  that  this  time  differs 
but  a  few  minutes  from  the  time  of  the  same  contact  for  the 
centre  of  the  earth,  and  during  these  few  minutes  we  may 
assume  the  distance  m  to  vary  uniformly. 

Let  T  be  the  time  of  the  geocentric  contact,  and  Tf  the 
required  time  of  the  contact  at  the  place,  both  times  being  reck- 
oned at  the  first  meridian.  At  the  time  T  the  geocentric  dis- 
tance =  s'  ±  s,  and  at  the  time  T1  the  apparent  distance 
mf  —  s' ±  s  (neglecting  here  the  augmentation  of  the  semi- 
diameters,  which  are  too  minute  to  be  considered  in  merely 
predicting  the  phenomenon) ;  but  at  this  time  T'  the  geocentric 
distance  has  become 


where  -  -  denotes  the  change  of  m  in  the  unit  of  time.     These 

(if 

values  substituted  in  (640)  give 

(Tr  —  T)~  =gf>  cos  A 

Differentiating  (636),  we  find 

dm    .     1/r  .    dM  .     ,r 

-  sin  M  4-  TJT-  m  cos  M  =  a  cos  <5.  =  n  sin  N 
dt  dt 

dm         ....       dM 

-T7-  cos  M m  sm  M  =  d  =  n  cos  J\ 

dt  dt 

whence 

dm  _ 

— -  =  n  cos  (M  —  JV) 

dt 

But,  since  at  the  time  T  we  have  m  =  sf  ±  s,  we  also  have  for 
this  time,  by  (638),  M  —  N  =  ^,  and,  therefore, 

dm 

which  gives 

'">COS*  (644) 


n  cos  4- 


in  which  the  values  of  n  and  ^  found  in  the  computation  for 
the  centre  of  the  earth  are  to  be  employed.  The  value  of  A  to 
be  employed  must*be  that  which  results  from  the  preceding 
formulae  at  the  time  T.  Now,  at  this  time  the  value  of  the  angle 


598  TRANSITS    OF    VENUS    AND    MERCURY. 

M  is  Q,  which  is  found  by  (638),  and  this  value  is  to  be  employed 
in  (642),  while  in  (641)  we  take  m  =  s'  ±  5. 
The  formula  for  T'  will  be 

T'  =  T  -f  -     —  [p  sin  <pf  sin  D  -f  />  cos  9?'  cos  D  cos  (0  —  w)1   (645) 

71  COS  ^ 

in  which  J1,  w,  i//,  Z>,  0,  TT  —  TT'  are  all  constants,  found  in  the 
computation  for  the  centre :  so  that  the  computation  for  a  par- 
ticular place  requires  only  this  single  formula  in  which  the 
latitude  and  longitude  of  the  place  are  to  be  substituted. 

358.  The  necessary  formulae  for  the  complete  prediction  are 
recapitulated  as  follows : 

I. FOR    THE    CENTRE    OF    THE    EARTH. 

Assume  a  convenient  time  T0  near  the  time  of  true  conjunc- 
tion of  the  sun  and  the  planet,  or  this  time  itself,  reckoned  at 
the  first  meridian,  and  find  for  this  time  the  values  of  a,  8  for 
the  planet;  a/,  S'  for  the  sun;  the  semidiameters  s  and  sf ; 
and  the  relative  changes  in  right  ascension  and  declination,  a 
and  (/,  in  the  unit  of  time.  Then,  putting  3Q  =  J  (d  +  o'),  compute 

m  sin  M  =  (a  —  a')  cos  d0  n  sin  N  =  a  cos  <?„ 

wcosJtf—    d  —  d'  n  cos  N  =  d 

m  sin  (M  —  N~] 


sin  4*  = 


6'' 


where  sf  -+*  s  is  to  be  employed  for  exterior  contact,  and  sf  —  s 
for  interior  contact.  Putting  h  =  3600,  to  reduce  the  terms  to 
seconds,  we  then  find 

T=T-h  cos  ^  -        cos    M  — 


in  which  cos  ^  is  to  be  taken  with  the  negative  sign  for  ingress 
and  with  the  positive  sign  for  egress. 

For  the  greatest  precision,  the  computation  may  be  repeated 
separately  for  ingress  and  egress,  taking  for  T0  the  value  of  T 
first  computed. 

As  in  solar  eclipses,  if  JJ  denotes  the  time  of  nearest  approach 
of  the  centres  of  the  bodies,  and  Al  the  distance  at  this  time,  we 
have 

Al  =  m  sin  (M  —  JVr)  Tv  =  T0  —  ^cos  (M  —  N) 


TRANSITS    OF    VENUS    AND    MERCURY.  599 

II.  —  CONSTANTS. 

For  each  of  the  computed  values  of  T  take  the  corresponding 
values  of  N  and  $  from  the  preceding  computation.     Then 

Q  =  N  -f  4 

Take  the  horizontal  parallaxes  x  and  n'  of  the  planet  and  the 
sun,  and  compute  A  and  D  by  the  formulae 

TT  4-  T:'  f  sin  F  =  sin  Y 

tan  r  =  —  !  --  tan  |  (sf  H-  s}  *        -n  s\ 

TT  —  TT  /  cos  .F  =  cos  r  cos  Q 

cos  J>  sin  (A  —  a0)  —       cos  f  sin  <2 
cosZ>  cos  (A  —  a0)  =  —  /sin  (<J0  -f  .F) 
sin  D  =      f  cos  (<50  -f  .F) 

in  which  a0  is  the  mean  of  the  right  ascensions  of  the  planet  and 
sun,  and  3Q  the  mean  of  their  declinations,  at  the  time  T. 

Find  the  sidereal  time  jj.  at  the  first  meridian  corresponding 
to  T.     Then  form  the  three  constants 


h  sin  D 


III.  —  FOR   A   GIVEN   PLACE  WHOSE   LATITUDE   IS   <p  AND   WEST 
LONGITUDE  O). 

Find  the  values  of  p  sin  <pf  and  p  cos  <p'  by  the  geodetic  table. 
The  required  time  of  the  phenomenon  at  the  place  is 

T'=  T  +  B  .  p  sin  <f  -f  C.  />  cos  <p'  cos  (0  —  w) 

The  local  time  will  be  T'—  co.  The  angle  Q  will  express  the 
angular  distance  of  the  point  of  contact  reckoned  on  the  sun's 
limb  from  its  north  point  towards  the  east,  and  will  be  very 
nearly  the  same  for  all  places  on  the  earth. 

EXAMPLE.  —  Compute  the  times  of  ingress  and  egress  for  the 
transit  of  Mercury,  November  11,  1861. 

I.  For  the  centre  of  the  earth.  —  Let  us  take  as  the  first  meridian 
that  of  Washington,  and  employ  the  elements  given  in  the 
American  Ephemeris. 

The  Washington  mean  time  of  conjunction  in  right  ascension 
is  November  11,  14*  59M  43\6,  which  we  shall  adopt  as  the  value- 
of  7.  For  this  time  we  have 


600 


TRANSITS    OF    VENUS    AND    MERCURY. 


a  =  a'=       227°  31' 


Hourly  motion  in  R.  A.  = 


Q 


«*•= 


17  32  45  .1 

17  44  44  .6 

17  38  44  .9 

0  11  59  .5 


3'    9".0 
"     =  +    2  32  .7 

g=  —    5  41  .7    —  -341"J 
in  Dec.  =  -f-    1  43  .8 
"        ==  —    0  40  .6 

d=  -f    2  24.4    =  +  144".4 


£  :r=        12".68 
Q  TT'=          8  .67 


0  Semidiameter  *'  = 


1  6'  1  2".  55 
4.94 


For  external  contacts,  «'  -f-  s  =        16  17  .49  = 

Since  for  T0  we  have  a  =  a',  we  also  have  M=  0°,  m  = 
=  719". 5.     We  then  find,  by  the  preceding  formulae, 
log  n  = 

M—N  = 

log  sin  4,  = 
For  Ingress,  4*  = 
For  Egress,  4  = 


977".  40 


2.55170 
66°    5M 
66     5.1 
9.82793 
137°42'.7 
42    17.3 


T0  =      14*  59W  43'.6 
N)=      —  49      7 .8 
Middle  of  Transit,  Tl=      14  10    35.8 
>s  4,  =  =p  2      1    48  .0 


Ingress,    T=      12      8   47 .8 
Egress,     T=     16    12   23 .8 
The  least  distance  of  the  centres  =  m  sin  (M  —  N)  =  10'  57".7 

II.   Constants. — We  find,  for  both  ingress  and  egress,  log  tan  f 
=  8.10094,  and  then  the  following  quantities : 


Ingress. 

Egress. 

Q 

71°  37'.6 

—  23°  47'.8 

log/ 

9.49891 

996252 

.F 

2°  17'.5 

0°  47'.3 

<S0 

17    40.3 

17    38.1 

5o+  F 

15    22.8 

16    50.8 

A  —  a0 

84    57.7 

-   56    16.0 

ao 

227    32.0 

237    30.8 

Ji 

312    29.7 

171    14.8 

log  sin  D 

9.48307 

9.94348 

log  cos  D 

9.97892 

9.68009 

T 

12*    8"47'.8 

16*  12W  23'.8 

Sid.  T.  Wash,  mean  noon 

15   23    17.8 

15   23    17.8 

1    59.7 

2    39.7 

ft 

3   34      5.3 

7   38    21.3 

JJL  (in  arc) 

53°  31'.3 

114°  35'.3 

fi  —  A  =  0 

101      1  .6 

303    20.5 

logS 

nl.2217 

1.6821 

log  C 

nl.7178 

1.4187 

OCCULTATION  OF  A  STAR  BY  A  PLANET.          601 

III.  For  any  place  on  the  surface  of  the  earth  we  have,  there- 
fore, in  mean  Washington  time, 

Ingress,  T'  =  12*    8m  47*.8  —  16'.66/>  sin  0'  —  52M9pcos  <j>'  cos  (101°    1'.6  —  u) 
Egress,    T'  —  16  12    23  .8  -j-  48  .10 p  sin  0'  -f  26  .23  p  cos  <j>'  cos  (303    20.5  — w) 

or,  in  a  more  convenient  form,  giving   the    logarithms  of  the 
constant  factors, 

Ingress,  7"  =  12*   8m47'.8  —  [1.2217]  p  sin  $'  +  [1.7176]  p  cos  $  cos  (w  -f-  78°  58'.4) 
Egress,    7"  =  16  12    23  .8  -f  [1.6821]  p  sin  0'  +  [1.4187]  p  cos  0'  cos  («  +  56   39.5) 

To  determine  whether  the  phenomenon  is  visible  at  the  given 
place,  we  have  only  to  determine  whether  the  sun  is  above  the 
horizon  at  the  computed  time.  All  the  places  at  which  it  will 
be  visible  will  be  readily  found  by  the  aid  of  an  artificial  terres- 
trial globe,  by  taking  that  point  wThere  the  sun  is  in  the  zenith 
at  the  time  T,  and  describing  a  great  circle  from  this  point  as  a 
pole.  All  places  within  the  hemisphere  containing  this  pole 
evidently  have  the  sun  above  the  horizon.  In  the  present 
example  this  point  at  ingress  is  in  latitude  —  17°  43'  and  longi- 
tude 186°  2'  west  from  Washington ;  and  at  egress  it  is  in  lati- 
tude --17°  46'  and  longitude  247°  4'.  The  whole  transit  is 
invisible  in  the  United  States,  and  in  Europe  only  the  egress  is 
visible. 

For  the  egress  at  Altona,  <p  =  53°  32'.8,  co  =  350°  3'.5,  we  find 

T'=       16*  13~  13-.0 

a,  =  -  .    5  47    57  .4 

Altona  mean  time  of  egress  =       22     1    10  .4 

The  time  actually  observed  by  PETERSEN  and  PAPE  was 
2%h  -jm  3^*  Tne  error  of  the  prediction  is  very  small,  and 
proves  the  excellence  of  LE  YERRIER'S  Theory  of  Mercury,  from 
which  the  places  in  the  American  Ephemeris  were  derived. 

OCCULTATION    OF    A    FIXED    STAR    BY   A    PLANET. 

359.  Very  small  stars  disappear  to  the  eye  when  near  the 
bright  limb  of  a  planet,  before  they  are  actually  occulted  by  it ; 
and  the  occultations  of  stars  of  sufficient  brightness  to  be  ob- 
served at  the  limb  of  the  planet  are  so  rare  that  it  has  not  been 
thought  worth  while  to  incur  the  labor  of  predicting  their  oc- 

*A*tron.  Nach.,  Vol.  LVI.  p.  239. 


602  .  PRECESSION. 

currence.  But  in  case  such  an  occultation  has  heen  observed  at 
different  points  on  the  earth,  it  may  be  reduced  by  Art.  341, 
substituting  the  planet  for  the  moon.  Such  observations  would 
be  especially  valuable  for  determining  the  planet's  parallax  by 
a  discussion  of  the  equations  of  condition  of  the  form  given  on 
p.  552.  If  the  occultation  occurred  near  the  stationary  points 
of  the  planet,  there  would  be  a  long  interval  between  the  im- 
mersion and  the  emersion;  the  coefficient  of  ATT  in  the  final 
equations  would  be  proportionally  large,  and  therefore  a  very 
accurate  determination  of  this  quantity  might  be  expected.  If, 
therefore,  means  can  be  found  to  make  the  occultation  of  the 
smaller  stars  by  a  planet  a  distinctly  observable  phenomenon, 
this  mode  of  finding  a  planet's  parallax  (and,  consequently,  also 
the  solar  parallax)  may  become  of  real  practical  value.* 

It  may  be  added  that  some  advantage  might  be  derived  from 
the  occultations  of  small  stars  by  the  dark  limb  of  Venus. 


CHAPTER  XL 

PRECESSION,    NUTATION,    ABERRATION,  AND   ANNUAL   PARALLAX 
OF   THE   FIXED    STARS. 

360.  I  HAVE  hitherto  treated  of  those  problems  only  in  which 
the  apparent  geocentric  places  of  the  celestial  bodies  are  sup- 
posed to  be  known ;  and  these  have  been  chiefly  problems  which 
may  be  regarded  as  arising  from  the  earth's  diurnal  motion,  or 
in  some  way  modified  by  it.  According  to  the  definition  of  our 
subject  (Art.  1),  Spherical  Astronomy  embraces  also  those  pro- 
blems which  arise  from  the  earth's  annual  motion  "so  far  as  this 
affects  the  apparent  positions  of  the  heavenly  bodies  upon  the 
celestial  sphere."  I  shall  therefore  proceed  now  to  consider 
those  uraiiographical  corrections,  affecting  the  apparent  geocen- 
tric places  of  the  stars,  which  result  from  the  motion  of  the 
earth  in  its  orbit,  and,  consequently,  also  those  which  result 


*  See  a  paper  by  PROF.  A.  C.  TWINING,  Enquiries  concerning  stellar  occultations  by 
the  moon  and  planets,  ^c.,  Am.  Journal  of  Science  for  July,  1858. 


PRECESSION.  603 

from  the  changes  in  the  position  of  the  plane  of  the  orbit  and 
the  plane  of  the  equator. 

361.  The   variations   of    astronomical   elements   are   usually 
divided  into  secular  and  periodic. 

Secular  variations  are  very  slow  changes,  which  proceed  through 
ages  (secula),  so  that  for  a  number  of  years,  or  even  centuries  in 
some  cases,  they  are  nearly  proportional  to  the  time. 

Periodic  variations  are  relatively  quick  changes,  which  oscillate 
between  their  extreme  values  in  so  short  a  period  that  they  can- 
not be  regarded  as  proportional  to  the  time  except  for  very  small 
intervals.* 

The  true  position  of  a  celestial  body,  or  of  a  celestial  plane,  at  a 
given  time,  is  that  which  it  actually  has  at  that  time ;  its  mean 
position  is  that  which  it  would  have  at  that  time  if  it  were  freed 
from  its  periodic  variations. 

362.  The  plane  of  the  ecliptic,  or  of  the  earth's  orbit,  is  a 
slowly  moving  plane.     Its  position  at  any  epoch,  as  the  begin- 
ning of  the  year  1800,  can  be  adopted  as  a  fixed  plane,  to  which 
its  position  at  any  other  time  may  be  referred. 

The  plane  of  the  equator  is  also  a  moving  plane.  Its  inclina- 
tion to  the  fixed  plane  and  the  direction  of  the  line  in  which  it 
intersects  that  plane  are  constantly  changing,  thus  causing 
variations  in  the  obliquity  of  the  ecliptic  and  in  the  position  of 
the  equinoctial  points. 

363.  The   latitudes   and   declinations   of  stars   are   therefore 
subject  to  variations  which  do  not  arise  from  the  motions  of  the 
stars,  but  from  the  shifting  of  the  planes  of  reference ;  and  the 
longitudes  and  right  ascensions  are  in  like  manner  subject  to 
variations  from  the   shifting  of  the  vernal   equinox,  which  is 
their  common  point  of  reference,  or  origin,  from  which  both  are 
reckoned. 

Under  the  head  of  precession  are  considered  those  parts  of 
these  variations  which  are  secular;  namely,  those  which  arise 
from  the  motions  of  the  mean  ecliptic  and  the  mean  equator. 

*  Most  of  the  secular  variations  also  have  periods,  though  of  great  length,  and 
therefore  not  yet  in  all  cases  well  denned:  so  that,  strictly  speaking,  the  distinction 
between  secular  and  periodic  variations  is  only  an  arbitrary  one,  established  for 
practical  convenience  between  variations  of  long  and  short  periods. 


604  PRECESSION. 

Under  the  head  of  natation  are  embraced  those  parts  of  these 
variations  which  are  periodic,  and  result  from  the  difference 
between  the  motions  of  the  true  ecliptic  and  equator  and  those 
of  the  mean  ecliptic  and  equator. 

PRECESSION. 

364.  Luni-solar  precession. — It  is  shown  in  physical  astronomy 
that  the  attraction  of  the  sun  and  moon  upon  that  portion  of 
the  matter  of  our  globe  which  is  accumulated  about  the  equator, 
and  by  which  its  figure  is  rendered  spheroidal,  combined  with 
the  rotation  of  the  earth  on  its  axis,  continually  shifts  the  posi- 
tion of  the  plane  of  the  equator  (without,  however,  changing  its 
inclination  to  the  plane  of  the  fixed  ecliptic).     The  line  of  the 
equinoxes,  or  the  intersection  of  the  two  planes,  is  thus  caused 
to  revolve  slowly  in  the   plane  of  the    ecliptic  in  a  direction 
opposite  to  that  in  which  longitudes  are  reckoned ;  the  result 
of  which  is  a  common  annual  increase  in  the  longitudes  of  all 
the  stars,  reckoned  on  the  fixed  ecliptic  by  a  quantity  which  is 
called  the  luni-solar  precession. 

The  luni-solar  precession  is,  then,  the  effect  of  a  motion  of 
the,  equator  upon  the  ecliptic. 

365.  Planetary  precession. — The  mutual  attraction  between  the 
planets  and  the  earth  tends  continually  to  draw  the  earth  out  of 
the  plane  in  which  it  is  revolving;  that  is,  to  change  the  position 
of  the  plane  of  the  orbit,  but  without  changing  the  position  of 
the  earth's  equator.     The  equator  here  being  regarded  as  fixed, 
and  the  ecliptic  as  moving,  the  effect  is  a  revolution  of  the  line 
of  intersection,  or  of  the  equinoxes,  in  the  plane  of  the  equator,  in 
a  direction  which  is  the  same  as  that  in  which  right  ascensions 
are  reckoned.     There  is  thus  caused  a  common  annual  decrease 
in   the   right   ascensions  of  all  the  stars,  which   is   called  the 
planetary  precession. 

The  planetary  precession  is,  then,  the  effect  of  a  motion  of  the 
ecliptic  upon  the  equator. 

366.  The  luni-solar  precession  does  not  affect  the  latitudes  of 
stars ;  but  since  it  changes  their  longitudes  it  must  also  change 
both   their   right   ascensions    and  declinations  (Art.  26).      The 
planetary  precession  does  not  affect  the  declination  of  stars,  but 


PRECESSION.  605 

changes    their    right    ascensions,    their    longitudes,    and    their 
latitudes  (Art.  23). 

367.  Obliquity  of  the  ecliptic. — Since  by  the  mutual  action  of  the 
planets  the  position  of  the  plane  of  the  (mean)  ecliptic  is  changed 
while  that  of  the  equator  remains  fixed,  the  mutual  inclination 
of  these  planes,  or  the  obliquity  of  the  ecliptic,  is  changed. 

The  action  of  the  sun  and  moon  in  causing  luni-solar  preces- 
sion does  not  directly  produce  any  change  in  the  obliquity  of 
the  ecliptic ;  but,  in  consequence  of  the  change  produced  by  the 
planets,  the  attraction  of  the  sun  and  moon  is  modified :  so  that 
there  results  an  additional  very  minute  change  of  the  inclination 
of  the  mean  equator  to  the  fixed  plane  of  reference. 

These  changes  produce  small  changes  in  the  co-ordinates  of 
the  stars,  which,  being  secular  in  their  character,  are  combined 
with  the  preceding  in  deducing  the  general  precession. 

368.  To  find  the  general  precession  in  longitude,  and  the  position  of 
the  mean  ecliptic,  at  a  gwen  time. — Let  NL,  Fig.  53,  be  the  fixed 
ecliptic,  or  the  mean  ecliptic  at  the 

beginning  of  the  year  1800;  AQ, 
the  mean  equator,  and  Fthe  mean 
vernal  equinox,  or,  as  it  is  briefly 
called,  the  mean  equinox,  of  1800. 
In  the  figure,  let  the  longitudes  be 
reckoned  from  V towards  N.  Let  Fa 

VVl  be  the  luni-solar  precession  in  longitude  in  the  time  t,  and 
AVQ  the  mean  equator  at  the  time  1800  -f-  t.  By  the  action  of 
the  planets,  the  ecliptic  in  the  same  time  is  moved  into  the  posi- 
tion NLV :  so  that  Vl  V.2  is  the  planetary  precession  in  the  time  t, 
and  V2  is  the  mean  equinox  at  the  time  1800  -f  t. 

The  point  N  may  be  called  the  ascending  node  of  the  mean 
ecliptic  on  the  fixed  ecliptic. 

The  difference  between  NV  and  NVZ  is  called  the  general 
precession  in  longitude,  being  that  part  of  the  change  of  the  longi- 
tudes of  the  stars  which  is  common  to  all  of  them. 

Now,  let  us  put 

e0  =  the  mean  obliquity  of  the  ecliptic  for  1800, 

=  NVQ, 
sl  =  the  obliquity  of  the  fixed  ecliptic  at  the  time  1800  -f  t, 

=  NV,  Q, 


606  PRECESSION. 

e   =  the  mean  obliquity  of  the  ecliptic  at  the  time  1800  4-  t, 

=  A'F,  Q, 
#  =  the  planetary  precession  in  the  interval  f, 

—  V  V 
ri  ** 

4  —  the  luni-solar  precession  in  the  interval  t, 

=  FF,, 
4/j  =  the  general  precession  in  the  interval  t, 


n  =  the  longitude  of  the  ascending  node  of  the  mean  ecliptic 
at  the  time  1800  -f-  1.  reckoned  on  the  fixed  ecliptic  from 
the  mean  equinox  of  1800, 


=  the  inclination  of  the  mean  ecliptic  to  the  fixed  ecliptic 
at  the  time  1800  -f-  f, 


The  first  five  of  these  quantities  will  be  here  assumed  as  known 
from  the  investigations  of  physical  astronomers.  The  following 
are  their  values,  according  to  STRUVE  and  PETERS,*  for  the  epoch 
1800: 

e0  =  23°  27'  54".22  \ 

e,  =  e0+  0".00000735£2  ) 

e  =£0—  0".4738£  —  (T.OOOOOUf1  >    (646) 

0  =  0".15119£  —  0".00024186Z2  \ 

4  ==  50".3798  1  —  0".0001084  1*  ] 

from  which  we  can  find  ^1?  II,  and  n,  as  follows.     In  the  triangle 
we  have 

#=FIF, 


180°-  e  =  JVFJr1  n  -f  4-  = 

and  hence,  by  the  GAUSSIAN  equations  [Sph.  Trig.  (44)] 

*  DR.  C.  A.  F.  PETERS,  Numerus  Constans  Nutationis,  pp.  66  et  71.  The  observa- 
tions at  Dorpat  give  0".4645  for  the  annual  diminution  of  the  obliquity,  and  this  is 
adopted  in  the  American  Ephemeris  instead  of  0".47o8,  which  results  from  theory 
and  is  subject  to  an  error  in  the  estimated  mass  of  Venus.  The  difference,  however, 
is  so  small  that  either  number  will  serve  to  represent  the  actually  observed  obliquity 
for  half  a  century  within  0".o. 

I  have  here  adopted  the  precession  constant  (50". 3798)  given  by  PETERS,  rather 
for  the  convenience  of  the  reader  (this  being  employed  in  the  English  and  American 
Almanacs)  than  on  account  of  its  superior  accuracy.  Recent  researches  rather 
confirm  BESSEL'S  constant  (50".36354).  See  MADLER'S  Die  Eigenbewegungen  der 
Fixstcrne,  Dorpat,  1856,  p.  11. 


PRECESSION.  607 


cos  J  TT  sin  J  (4  —  40  =  sin  J  #  cos  £  (e  -f-  e.) 

COS  J  7T  COS  *  (4  —  ty)  =  COS  J  *  COS  J  (e  —  f$ 

Bin  JTT  sin  (n  -|-  i  4  -}-  i  4i)  =  8^n  i *  8^n  i  (e  +  ei) 

sin  £;r  cos  (n  -j-  J  4  -f-  i  4j  =  cos  J*  sin  J  (e  —  et) 


(647) 


The  angles  J  &  and  J  (e  —  ej  are  so  small  that  their  cosines  may 
always  be  put  equal  to  unity,  and,  consequently,  also  those  of 
JTT  and  J(i//  —  ^)  ;  while  for  their  sines  we  may  substitute  the 
arcs.  We  thus  obtain  at  once,  from  the  first  two  equations, 

4  —  *!  =  '9  cos  J(e  +  O 
where  we  can  take,  with  sufficient  accuracy, 

COS  J  (e  +  e,)  =  COS  (e,  —  0".2369f) 

==  cos  e0  -f-  0.".2369^  sin  1"  sin  s0 
and  hence,  by  substituting  the  values  of  $  and  e0  from  (646), 


—  0".0002218<2 
4,  =  50".2411#  +  0".0001134^2  (648) 

The  sum  of  the  squares  of  the  last  two  equations  of  (647)  gives 

**  =  W  sin2  J  (e  +  ej  +  («  —  e,)2 
in  which  we  may  take 

sin'  \  (e  +  e,)  =  sin2  s0  —  0^.2369  1  sin  1"  sin  2e0 
and  then,  substituting  the  values  of  $,  e0,  and  s  —  sv  we  obtain 

7r2r=:  0".228111  1*  —  0".0000033234  1» 
and,  by  extracting  the  root, 

TT  =  Q"A776t  —  0".0000035f2  (649) 

The  quotient  of  the  third  equation  of  (647)  divided  by  the 
fourth  gives 

tan  (n  -f  J  4  -f  **!)  =  -      -  sin  }(e  -f  e,) 

"  ei 

in  which  we  have 

*  0.15119  1  —  0.00024186  ** 

e  _  ^  ~     _  0.4738  1  —  0.0000875  1> 

=  —  0.3191  -f-  0.00051636* 


608  PRECESSION. 

and 

sin  J  (s  -f  £:)  =  sin  e0  —  0".2369£  sin  1"  cos  e0 
whence 

tan  (n  -f-  H  4-  Hx)  =  —  0.127062  -f-  0.00020595  # 

If,  then,  we  put 

tan  n0=  —  0.127062 
or 

nQ=       172°  45' 31" 
and  also 

we  have 

tan  nx  —  tan  n0  =  (nx  —  n0)  sin  1"  sec2  n0  ==  0.00020595  t 


sin  A 
whence 

n1  =  n  +  H  +  H1=  1T2°  45'  31"  +  41".  805  1 
and,  subtracting  from  this  the  quantity 


1 
we  have,  finally, 

n  =  172°  45'  31"  —  8".505£  (6bO) 

The  equation  (648)  determines  the  general  precession,  and  (649) 
and  (650)  the  position  of  the  mean  ecliptic. 

369.  To  find  the  precession  in  longitude  and  latitude  of  a  given  star, 
from  the  epoch  1800.—  Let  LNB  (Fig.  54)  be  the  fixed  ecliptic 
of  1800;  LlNBl  the  mean  ecliptic  at  the  given  time  1800  -f-  1;  P 
and  Pl  the  poles  of  these  circles  respectively.  The  node  N  is 
the  pole  of  the  great  circle  PP^  joining  P  and  Pv.  Let  S  be 
the  star,  and  put 

•  L  =  the  star's  given  mean  longi- 
tude for  1800,  reckoned  from 
the  mean  equinox  of  that 
year, 

B  =  the  star's  given  mean   lati- 
tude  for  1800, 

A,  ft  =  the  mean  longitude  and  lati- 
tude for  1800  -f  t. 

We  have  in  the  figure  (as  in  Fig.  53) 


=  L  —  n 
=  ^   —FT— 


PRECESSION.  609 

and  in  the  triangle  PSP^  we  have 


PPl  =  LLV  = 

PS  =90°  —  B 

PiS=W  —  p 
SPPl  =  EL  =  90°  +  L  —  n 

SPl  P  =  180°  —  L&  =  180°  —  (90°  -f  ;  _  n  —  4t) 
=  90°  -  (A  -  n  -  +1) 

so  that,  by  the  fundamental  equations  of  Sph.  Trig., 

cos  /9  cos  (A  —  n  —  40  —     cosJ?cos(£  —  n)  ^ 

cos  /9  sin  (A  —  n  —  4,)  =      cos^sin  (L  —  n)cos7r-}-  sin  B  sin  K  V  (651) 
sin  /?  =  —  cos  5  sin  (Z-  —  n)sin7r-f-  sin  B  cos?:) 

Instead  of  these  rigorous  formulae,  we  may  deduce  approximate 
ones,  which  will  be  sufficient  in  all  practical  cases,  as  follows. 
Neglecting  the  square  of  TT  (that  is,  putting  cos  n  =  1),  let  the 
first  equation  be  multiplied  by  sin  (L  —  II),  the  second  by  cos 
(L  —  II)  ;  the  difference  of  the  products  is 

cos  /3  sin  (A  —  L  —  4^)  =  sin  TT  sin  B  cos  (L  —  n) 

The  sum  of  the  products  obtained  by  multiplying  the  same 
equations  by  cos  (L  —  II)  and  sin  (L  —  II),  respectively,  is 

cos  /?  cos  (A  —  L  —  4,)  =  cos  B  -f-  sin  TT  sin  B  sin  (L  —  n) 
and  the  quotient  of  these  last  equations  is 

tan  (A  -  L  -  4,)  =      "in  «  tai»Bco»C&  -  n) 

1  -f  sin  TT  tan  B  sin  (L  —  n) 

which  developed  in  series  (PL  Trig.,  Art.  257)  gives 

A  —  L  —  ^  —  TT  tan  B  cos  (L  —  n)  —  J  **  tan2  B  sin  2  (Z  —  n)  —  &c. 

where,  however,  since  we  here  neglect  the  square  of  JT,  the  first 
term  of  the  series  suffices  :  so  that  we  have 

X  —  L  —  4t  -f-  TT  tan  B  cos  (L  -^  n)  (652) 

Here  ^  appears  as  the  precession  in  longitude  common  to  all 
the  stars,  and  the  term  n  tan  B  cos  (L  —  II)  as  that  which  varies 
with  the  star. 

The  last  equation  of  (651)  gives 
/ 

sin  /3  —  sin  B  =  —  sin  ?r  cos  B  sin  (L  —  n) 
VOL.    I.—  39 


610  PRECESSION. 

whence,  neglecting  7?  as  before, 

/?  —  B  =  —  v  sin  (L  —  n)  (653) 

The  values  of  ^,  TT,  and  II  being  found  for  the  time  1800  -f-  /, 
by  means  of  (648),  (649),  and  (650),  the  formulae  (652)  and  (653) 
determine  the  required  precession  in  the  longitude  and  latitude, 
and,  consequently,  also  the  mean  place  of  the  star  for  the  given 
date. 

370.  To  find  the  precession  in  longitude  and  latitude  between  any  two 
given  dates.  —  Suppose  /  and  ft  are  given  for  1800  +  t,  and  /'  and  /9' 
are  required  for  1800  -j-  t'.  Denoting  by  L  and  B  the  longitude 
and  latitude  for  1800,  we  shall  have,  by  (652), 

A  —  L  =  4/j  -f-  TT  tan  B  cos(L  —  n  ) 
/  —  L  —  4/  -f  r'tan  B  cos(£  —  n') 

where  ^/,  ;r',  II'  are  the  quantities  given  by  (648),  (649),  and 
(650)  when  tf  is  substituted  for  L  If  we  subtract  the  first  of 
these  equations  from  the  second,  and  at  the  same  time  introduce 
the  auxiliaries  a  and  A,  determined  by  the  conditions 

a  sin  A  =  (d  -f-  ?r)  sin  £  (n'  —  n) 

a  COS  A  —  (TT'  —  TT)  COS  £  (n'  —  H) 

we  find 

A'  —  >l  =  4,'—  4  +  a  COS/.L  -     mp?     - 

and  in  the  same  manner,  from  (653), 


For  the  values  of  A  and  a  we  have 

tan  A  =  ^±_?  tan  l(n'  —  n)  =  ^  4tan  *(n'—  n) 

7T    -  7T  t    -  C 

or,  by  (650), 


so  that  cos  A  may  be  put  equal  to  unity,  and  therefore  we  have 

a  =  nf  —  * 


PRECESSION.  611 

We  may  also  put  tan  ,9  instead  of  tan  I?  in  the  above  formulae, 
since  the  error  in  )J  —  X  thus  produced  will  be  only  a  term  in  7? ; 
and  for  L  we  may  take  I  —  ^  :  so  that  if  we  put 

n'-fn 
Li A  =  A  —  M 


and  then  substitute  the  numerical  values  of  our  constants,  we 
shall  have  the  following  formulae  for  computing  the  precession 
from  1800  +  t  to  1800  +  tf : 

M=     172°  45'  31"  -f  t .  50".241  —  (f  -f  Q  8".505 
A'— X=     (t'—t)  [50".2411  +  (*'-K)0".0001134] 

_j_  (f — f)  [0".4776  —  (tf  -f  0  0".0000035]  cos  (A— M )tan/9 
fi'-.p  =  —  (f—t)  [0".4776  —  (tf+  f)0"-0000035] sin  (^— -M") 

These  are  the  same  as  BESSEL'S  formulae  in  the  Tabulae  Regiom,on- 
tance,  except  that  we  have  here  employed  the  constants  given  by 
PETERS,  and  the  epoch  to  which  t  and  t1  are  referred  is  1800. 

To  find  the  annual  precession  in  longitude  for  a  given  date. — If  we 
divide  the  equations  (654)  by  t'  —  t,  the  quotients 


(654) 


t'—t'  f  —  t 

will  express  the  mean  annual  precession  between  the  two  dates ; 
and  if  we  then  suppose  t'  and  t  to  differ  by  an  infinitesimal 
quantity,  or  put  t'  =  t,  these  quotients  will  become  the  differen- 
tial coefficients  which  express  the  annual  precession  for  the  in- 
stant 1800  -f-  t ;  namely, 


'-  =       50".2411  +  0".0002268f 

+  [0".4776  —  0.0000070  £]  cos  (A  —  M)  tan 

~  =  —  [0".4776  —  0.0000070  f|  sin  (A  —  M) 
dt 

in  which 

M  =       172°  45'  31"  -j-  33".23 1 


(655) 


EXAMPLE. — For  the  star  Spica,  we  have,  for  the  beginning  of 
the  year  1800, 

the  mean  longitude,  L  =       201°  3'    5".97 
the  mean  latitude,      B  =  —      2°  2'  22".64 


612  PRECESSION. 

Find  its  mean  longitude  and  latitude  for  the  beginning  of  the 
year  1860. 

First.  By  the  direct  formulae  (652)  and  (653).— "We  find,  by 
(648),  (649),  and  (650),  for  t  =  60, 

4j  =  50'  14".874 
«  =  28".6434 
n  =  172°  37'  1" 
whence 

L  —  n  =  28°  26'  5" 

T:  tan  J5  cos  (L  —  n)  =  —    0".897 

TT  sin  (L  —  n)  =  -j-  13".639 

and  hence,  by  (652)  and  (653),  the  precession  is 

X  —  L  =       50'  14".874  —  0".897  =  50'  13".977 
/5  —  B  =  —  13".639 

and  the  mean  longitude  and  latitude  for  1860.0  are 

X  =       201°  53'  20".95 
p=-       2°    2'36".28 

Second.  By  the  use  of  the  annual  precession. — The  mean 
annual  precession  for  the  sixty  years  from  1800  to  1860  is  the 
annual  precession  for  1830.  Hence,  by  taking  t  =  30  in  (655), 
and  denoting  by  ^  and  /90  the  longitude  and  latitude  for  1830, 

j  =       50".2479  -f  0".4774  cos  (/10  —  M )  tan  /?0 

~=  —  0".4774  sin  (J10—  Jf) 
M  =       173°  2'  8". 

To  compute  these,  we  can  employ  approximate  values  of  >*0  and 
/90,  found  by  adding  the  general  precession  for  thirty  years  to  .L, 
and  neglecting  the  terms  in  TT  ;  namely, 

;0  =  201°  28'.2  ft  =  —  2°  2'.6 

and  hence  /0  —  M  =  28°  26'.1, 

f  =  50".2329  4?  -=  —  0".2274 

at  at 

These  multiplied  by  60  give  the  whole  precession  from  1800  to 
1860, 

/I  —  L  =  50'  13".97          /?  —  B  =  —  13".64 

agreeing  with  the  values  found  above. 


PRECESSION. 


613 


Fig.  55. 


371.  Given  the  mean  right  ascension  and  declination  of  a  star  for 
any  date  1800  +  £,  to  find  the  mean  right  ascension  and,  declination  for 
any  other  date  1800  +  t'.—  Let  Vl  F/ 
(Fig.  55)  be  the  fixed  ecliptic  of 
1800,  Vj§  the  mean  equator  of 
1800  +  t,  Vi'Q  the  mean  equator 
of  1800  +  /',  Q  the  intersection  of 
these  circles  (or  the  ascending  node 
of  the  second  upon  the  first).  The 
position  of  the  point  Q  is  found  as 
follows.  The  arc  Vl  Vj'  is  the  luni-solar  precession  for  the  in- 
terval tf  —  t  :  so  that,  distinguishing  by  accents  the  quantities 
obtained  by  (646)  when  t'  is  put  for  £,  we  have,  in  the  triangle 


r.r;  =+'-+, 

and  putting 

QYi==9Q°  —  z,  §F/=90°+^,  Fj 

we  find,  by  GAUSS'S  equations  of  Sph.  Trig., 

cos  2  0  sin  \  (z'  -}-  z)  =--  sin  £  (4'  —  4)  cos  J-  (e/  -f- 

COS  5  0  COS  2  (2'  -}-*)  =  COS  2  (4'  —  4)  COS  2  (e/  — 

sin  20  sin  2  (2'  —  z)  =  cos  £  (4'  —  4)  sin  J  (c^  — : 
sin  1 0  cos  2  (2'  —  2)  =  sin  2  (4'  —  4)  sin  J  (e/  -}- 

which  determine  0,  2,  and  2'  in  a  rigorous  manner.  But,  since 
J  (e/  —  ej)  is  exceedingly  small,  we  can  always  put  unity  for  its 
cosine,  and  the  arc  for  the  sine,  and,  consequently,  the  same 
may  be  done  in  the  case  of  the  arc  J  (zf  —  z) ;  we  thus  obtain 
the  following  simple  but  accurate  formulae  : 


(656) 


tan  *  (zr  +  z)  =  tan  *  (4'  —  4)  cos  J  (e/ 


*(«/-',) 


K*'-*)- 


sin  |0  =  sin  J  (4'  —  4)  sin  J  (e/  -f- 


(657) 


If  T^  and  T^'  are  the  positions  of  the  mean  equinox  in  1800  -f  t 
and  1800  -f-  *',  Vi  F2  is  the  planetary  precession  for  the  first  and 
Vi  V2'  that  for  the  second  of  these  times,  which  being  denoted 
by  &  and  #'  we  have 

V3  Q  ===  90°  —  2  —  $ 
V'     =  90°      2'-  *' 


PRECESSION. 


If  then  we  put 

0,,  (5  =  the  given  mean  right  ascension  and  declination  of  a 

star  S,  for  1800  -f  f, 
a',«'=  those  required  for  1800  -f  f, 

we  have  a  =  V2D,  and  a'  =  V2fD,  and,  consequently, 

QD  =  F9D  —  F,  C  =  a  -f  z  +  *  -  90°, 
F/D'  —  F2'  Q  =  a,'  —  z'  +  ft'  —  90°, 


Now,  let  P  and  P'  (Fig.  56)  be  the 
poles    of    the    equator   at   the    times 
]D    1800  -f  t,    1800  +  t',    AQD,    A'QD', 
the  two  positions  of  the   equator  at 
these  times,  as  in  Fig.  55 ;  S  the  star. 
M§  is  the  pole  of  the  great  circle  PP'A' 
joining    the    poles   P  and   P',    and, 
therefore,  PP'  =  AAf  =  AQA'  =  0, 
and  in  the  triangle  PP'S  we  have 

PS=9Q<>—  3,  P'S=  90°  —  S',  PP'=Q 

SPP'=  AD=  90°  -f  QD  =  a  -f  z  -f  ft 

SP'P=  180°  —  A'D'=  90°  -  Qiy  =  180°  —  (a'  —  z'  -f  0') 

Hence,  by  the  fundamental  equations  of  Spherical  Trigonometry, 

cos  S'  sin  (a' — 2' -(-#')  =  cos  fl sin  (a  -f-2-j-tf)  "j 

cos  d'  cos  (a! — 2:'-{-y)  =  cos^cos(a -f  2:-j-»9)cos0 — sin<5sin0    V   (658) 
sin  «5'=cos5cos  (a  -\- z  -j-»9)sin0- 


We  have  thus  a  rigorous  and  direct  solution  of  oiy*  problem  by 
finding,  first,  0,  z,  and  z'  from  (656),  and  hence  a'  and  o'  by  (658), 
employing  the  values  of  e,  oj/,  $  for  the  time  1800  -f  /,  and  of 
ex,  V,  V  for  the  time  1800  +  *',  as  given  by  (646)  for  the  two 
dates. 

372.  The  formulae  (658)  may  be  adapted  for  logarithmic  com- 
putation by  the  introduction  of  an  auxiliary  angle  in  the  usual 
manner;  or  we  may  employ  the  GAUSSIAN  equations,  which,  if 
we  denote  the  angle  at  the  star  by  C,  and  for  the  sake  of  brevity 
put 

A  =  a  +  z  -f  ft  A'  =  a'  —  z'  +  ft'  (659) 


PRECESSION.  615 

give 

cos  }  (90°  +  <$')  sin  i  (A'+C)  =  cos  }  (90°  -f  *  —  8)  sin  M 
cos  J  (90°  H-  <S')  cos  *  (J/+<7)  rrr  cos  i  (90°  -{-  <5  -f-  0)  cos  M 
sin  £  (90°  -f  5')  sin  £  (^'—(7;  =  sin  £  (90°  -f-  <J  —  0)  sin  M 
sin  J  (90°  -f  *')  cos  £  (A'—  (7)  =  sin  J  (90°  -f  <S  -f  8)  cos  \A 

373.  We  may,  however,  obtain  greater  precision  by  computing 
the  differences  between  A  and  A'  and  between  d  and  d'.  From 
the  first  two  equations  of  (658)  we  deduce 

cos  df  sin  (Af — A)  =  cos  d  sin  A  sin  0  [tan  d  -f-  tan  £0  cos  A] 

cos  d'  cos  (J/ — A)  =  cos  <5  —  cos  d  cos  J.  sin  0  [tan  d  -f-  tan  j  0  cos  J.] 

so  that,  if  we  put 

p  =  sin  0  (tan  d  -f-  tan  £  0  cos  A) 


we  have 


1  — pcos  A 
and,  by  NAPIER'S  Analogy,* 


(660) 


EXAMPLE. — The  mean  place  of  Polaris  for  1755,  according  to 
the  Tabulae  Regiomontance,  is 

a  =  10°  55'  44".955  d  ==  87°  59'  41".12 

it   is   required  to  reduce  this  place  to  the  mean  equator  and 
equinox  of  1820. 

For  1755  we  take  t  =  —  45 ;  and  for  1820,  t'  =  -f-  20  ;  and,  by 
(646),  we  find— 

For  1755.  For  1820. 

*  =  —  37'  47".31  4/=  +  16'  47".55 

#  =  -    7".29  #=         -f    2".93 

e,  =  23°  27'  54".23488  e/=  23°  27'  54".22294 

and  hence 

i-  (V  —  4)  =  27'  17".43 
^(£i'_ei)  ==  _      0".00597 
}  (£'  -f  e)   =  23°  27'  54".23 

*  The  formulae  (657),  (658),  (659),  (660)  are  those  given  by  BESSEL  in  the  Tabulss 
Regiomontanse. 


616  PRECESSION. 

with  which  the  formulae  (657)  give 


)  =  -|-  25'  2".02 
$(z'—z}=         -  1".89 
z  =  -f  25'  3".91 
z'=  -f  25'0".13 
log  sin  1  0=7.499823 

Then,  by  the  formulae  (660),  we  find 

A  =  a  -f  z  -f  $  =  11°  20'  41".57 

log  ;?  9.256676  log  tan  \  0  7.499825 

log  sin  JL  9.293836  log  cos  \  (A'+A)  9.989446 

log  cos  J.  9.991430  log  sec  J  (A'—  A)  0.000101 

log^cos^t  9.248106  log  tan  J  (5'  —  5)    9.489372 

ar.  co.  log  (1  —  p  cos  A)  0.084629 

log  tan  (A'  —  A)  8.635141 

A'—  A=  2°28'18".08  df  —  8=         21'  12".99 

A'  =  13°  48'  59".65 
—  0'=14°  13'  56".85  3'  =  88°  20'  54".ll 


374.  To  find  the  annual  precession  in  right  ascension  and  declina- 
tion. —  In  computing  the  precession  for  a  single  year,  the  square 
of  0  becomes  insensible,  and  we  may  take,  instead  of  (660),  the 
approximate  formula 

A'—  A  =  a'  —  a  —  (zr  -f  z)  -f  #'  —  #  =  0  sin  a  tan  d 
and  from  (657)  wo  then  have,  with  sufficient  accuracy, 

Zf  +  2  =  (4,'  —  4)  COS  e, 
0  =  (4,'  —  4)  sin  et 

Substituting  these  values  in  the  above  formula,  and  then  dividing 
by  f  —  /,  we  have 

'  #'—  *    ,   V—  4    . 

cos  e«  ---  --  sin  £,  sin  a  tan  o 
'' 


« 
t'  —  t        f—t  t'—tt'  —  t 

which  gives  the  annual  precession  between  the  times  1800  -f  t  and 
1800  +  t',  the  unit  of  time  being  one  year.  But,  in  order  that 
the  formula  may  express  the  rate  of  change  at  the  instant 
1800  +  t,  we  must  suppose  the  interval  f  —  t  to  become  infinitely 
small  ;  that  is,  we  must  write  the  formula  thus  : 


PRECESSION. 


617 


da,        d4  dft       d-± 

-  =  — -  cos  e, sin  e  sin  a  tan  d 

dt         dt  dt         dt 

and  similarly,  from  the  last  equation  of  (660), 
dS 


Putting  then 


we  find,  by  (646), 


=  —  sin  e,  COS  a 
dt        dt          l 


m  =  —  cos  e,  —  — 
dt  dt 


n  =  —  sin  e. 
dt 


(661) 


and  hence 


—  cos  e,  =(50".3798— 0".0002168Q  cos 

=  46".2135  —  0".00019887 1 
—  =    0".1512  —  0".00048372 1 

m  =  46".0623  -f  0".0002849 1 
n  =  20".0607  —  0".0000863 1 


}    (662) 


and  the  annual  precession  in  right  ascension  and  declination  for 
the  time  1800  -f  t  is  found  by  the  formulae 


—  =  m  4-  n  sin  a  tan  3 
dt 

dd 


dt 


=  n  cos  a 


(663) 


These  formulae  may  be  used  for  computing  the  whole  precession 
between  any  two  dates,  if  we  multiply  the  annual  precession  at 
the  middle  time  between  the  two  dates  by  the  number  of  years  in 
the  interval. 

EXAMPLE. — The  mean  right  ascension  and  declination  of  Spica 
for  1800  are,  by  the  Tabulce  Regiomontaruz, 

a  =       13*  14*  40-.5057 
d  =  —  10°    6'  46".843 

Find  the  mean  right  ascension  (a')  and  declination  (<?')  for  1860. 
We  have,  for  1830,  by  making  t  =  30  in  (662), 


m  =  46".0708 


n  =  20".0581 


618  PRECESSION. 

and,  for  a  first  approximation,  taking  af  =  a,  3'  =  £,  we  have,  by 
(663), 

§=  +  47".22  £  =  - 19-.00 

The  approximate  precession  for  sixty  years  is,  therefore, 

in  R  A.,  -f  2833"  =  -f  188-.9  in  dec.,  —  1140" 

which,  applied  to  a  and  £,  give  the  approximate  values  for  1860, 

a'  =  13*  17-  49'.4  d'  =  —  10°  25'  47" 

The  means  between  these  values  and  those  of  a  and  S  are 
a0  =  13*  16-  15'.  30  =  —  10°  16'  17" 

which  being  employed  in  (663)   give  the  more  correct  annual 
precession  for  1830, 

fk  =  +  47".2579  4r  =  —  18".9582 

at  at 

The  true  precession  for  sixty  years  is  then 

in  E.  A.,  +  2835".474  =  3-  9-.0316,  in  dec.,  —  18'  57".492, 

which  applied  to  a  and  d  give  the  mean  place  for  1860, 

a'  =  13*  17"  40-.5373  d'  =  —  10°  25'  44".335 

and   these  values   agree  almost  precisely  with  those  found  by 
the  rigorous  method  of  Art.  371. 

375.  To  find  the  position  of  the  pole  of  the  equator  at  a  given  time. — 
The  precession  causes  the  pole  of  the  equator  to  Devolve  about 
the  pole  of  the  ecliptic  (nearly)  in  a  small  circle  whose  polar 
distance  is  equal  to  the  obliquity  of  the  ecliptic.  The  time  in 
which  the  pole  will  make  a  complete  revolution  and  return  to 
the  same  position  (small  changes  in  the  obliquity  of  the  ecliptic 
not  considered)  is  the  value  of  t  given  by  the  equation 

50".241U  +  0".0001134*2=  360°  X  60  X  60  —  1296000" 

which  gives 

t  =  24447  years; 

or,  in  round  numbers,  since  the  precession  is  not  known  with 


PRECESSION.  619 

sufficient  precision  to  determine  so  great  a  period  exactly, 
t  =  24500  years. 

To  find  the  position  of  the  pole  for  any  indeterminate  time 
1800  -j-  tf,  we  have  only  to  observe  that  if  P,  in  Fig.  56,  is  the 
pole  for  a  fixed  time  1800  +  t,  P'  that  for  the  time  1800  -f-  *', 
the  right  ascension  of  P',  reckoned  from  the  equinox  of  1800  -f  /, 
is  equal  to  that  of  the  point  Q  diminished  by  90°.  The  right 
ascension  of  §  is  V2Q  in  Fig.  55,  and,  in  Art.  871,  we  have  found 

VaQ  =  90°  — 3  — rt 
Hence,  if  we  put 

A,  D  =  the  right  ascension  and  declination  of  the  pole  at  the 
time  1800  -f-  f,  referred  to  the  equator  and  equinox 
of  1800  -f  t, 
we  have 

A  =  —  z  —  4 
D=90°  — 0 

which  will  become  known  by  computing  ^>  ^',  £>  s1 '•>  &  for  the 
times  1800  +  /,  1800  +  t',  and  then  z  and  0  by  (657) 

An  approximate  solution  is  obtained  by  neglecting  the  varia- 
tion of  e,  and,  consequently,  taking  zf  =  z,  and  also  neglecting  # : 
so  that 

tan,i  =  —  tani(4/—  4)  cos  e0  1 

sin  (45°  —  JD)  =       sin  J  (V  —  4)  sin  e0  j     ^ 

The  ambiguity  in  determining  A  by  its  tangent  is  removed  by 
observing  that  cos  A  and  cos  J  (-\J/  —  -J/)  must  have  the  same  sign 
so  long  as  ^'  —  ^  does  not  exceed  360°,  as  we  readily  infer  from 
the  equations  (656). 

For  example,  if  we  wish  to  find  the  position  of  the  pole  for 
the  year  14000,  referred  to  the  equinox  of  1850,  we  take  t  —  50, 
t'=  12200  ;  whence  ^  —  $  =  165°  33',  and 

A  =  277°  52'  D  =  43°  28' 

The  position  of  a  Lyrce  for  1850  is 

a  =  277°  58'  8  =  38°  39' 

consequently,  this  star,  in  the  year  14000,  will  be  within  five 
degrees  of  the  pole,  and  will  become  the  pole  star  of  that  period. 


620  PROPER    MOTIOX    OF    THE    FIXED    STARS. 


PROPER    MOTION    OF   THE   FIXED    STARS. 

376.  "When  from  direct  observations  of  the  apparent  positions 
of  the  stars  we  deduce  their  mean  places,  we  find  that  the  changes 
in  these  mean  places  between  distant  dates  do  not  agree  with 
those  which  arise  solely  from  the  precession,  but  that  each  star 
appears  to  have  a  small  motion  of  its  own,  which  is,  therefore, 
designated  as  its  proper  motion.* 

This  proper  motion  is  partly  real — arising  from  the  absolute 
motion  of  the  star  in  space ;  and  partly  apparent — arising  from 
the  motion  of  our  own  sun,  with  the  planets,  whereby  our  point 
of  view  is  changed.  It  will  be  shown  hereafter  how  these  two 
motions  are  to  be  distinguished  from  each  other;  but  we  here 

O 

consider  only  the  resultant  of  both. 

The  path  of  a  star  upon  the  celestial  sphere  is  assumed  to 
coincide  with  the  arc  of  a  great  circle,  and  the  proper  motion  in 
this  circle  to  be  uniform  or  proportional  to  the  time.  It  is  not 
probable  that  either  hypothesis  is  strictly  true ;  but  that  portion 
of  its  whole  orbit  which  a  star  appears  to  describe  even  in  several 
centuries  is  so  small  that,  in  the  observations  thus  far  practicable, 
no  sensible  departure  from  uniform  motion  or  from  motion  in  a 
great  circle  could  become  sensible. 

377.  In  order  to  distinguish  the  proper  motion  from  the  pre- 
cession, the  star's  observed  mean  place  at  two  different  dates 
must  be  referred  to  the  same  mean  equinox.    Suppose,  therefore, 
that  a  and  3  are  the  observed  mean  right  ascension  and  declina- 
tion for  the  time  1800  -j-  t,  and  a'  and  V  those  for  1800  +  t> '.    If 
we  start  from  the  first  place,  and,  computing  the  precession  for 
the  interval  t'  —  t,  find  the  values  (a')  and  (3')  for  1800  +  *',  the 
whole  proper  motion  in  the  interval,  referred  loathe  equinox  of 
1800  +  tf,  is 

Aa'  ==  a'  —  (a')  A<J'  =  #  —  (£') 

But  if  we  start  from  the  second  place,  and,  reducing  it  to  the 
first  time,  find  (a)  and  (d\  the  proper  motion  in  the  interval, 
referred  to  the  equinox  of  1800  -f-  /,  is 

Aa  =  (a)  —  a  Ad  —  (<5)  —  3 

*  The  student  must  remember  that  precession  does  not  affect  the  relative  positions 
of  the  stars,  but  only  shifts  the  circles  of  reference.  The  proper  motion  changes  the 
relative  positions  or  the  apparent  configuration  of  the  stars. 


PROPER    MOTION    OF   THE    FIXED    STARS.  621 

378.  To  reduce  a  star's  mean  place  from  one  epoch  to  another,  when 
the  proper  motion  is  given. — Let  a,  £,  be  the  given  place  for  1800  -+-  /, 
and  let  the  given  annual  proper  motion  in  right  ascension  and 
declination,  referred  to  the  equinox  of  this  date,  be  denoted  by 
da  and  dd.     To  reduce  to  the  date  1800  -f  f,  we  first  find  the 
whole  proper  motion  in  the  interval,  by  the  formulae 

Aa  =  do,  (f  —  0  A<5  =  dd  (f  —  0 

Then,  putting 

(a)  =  a  -f  Aa  (<S)  =  d  -f-  A<S 

we  compute  the  precession  by  the  formulae  of  Arts.  371  to  374, 
employing  in  these  formulae  (a)  and  (d)  for  a  and  d. 

If  the  proper  'motion  (Aa',  A*?')  had  been  given  for  the  epoch 
1800  -f  f,  we  should  first  have  computed  the  precession  with  the 
given  values  a  and  d,  and,  having  applied  it,  if  (a')  and  (d'}  were 
the  resulting  values,  we  should  have  finally  a'  =  (a')  -f  Aa', 
d'=(dr)  +  AO'. 

379.  To  reduce  the  proper  motion  in  right  ascension  and  decimation 
from  one  epoch  to  another. — If,  in  Fig.  56,  P  and  Pf  are  the  poles 
of  the  equator  for  the  epochs  1800  -f-  t  and  1800  -f-  /'  respectively, 
and  we  suppose  the  star  S  to  vary  its  position,  the  present  problem 
requires  us  to  deduce  the  relations  between  the  variations  of  the 
parts  of  the  triangle  SPP',  the  side  PPf  being  the  only  constant 
part.     Observing  the  notation  of  Art.  371,  we  have  (since  £,  #, 
£',  #'  do  not  depend  upon  the  star's  place) 

d(SPP')  =  d(o  +  C  +  0)  =  da 
d(SP'P)  =  d(180°  —  «'  -f  C'  —  »»')  =  —  da,' 
d(SP}  =  —  dd 
=  —  dd' 


and  hence,  by  the  formulae  (47)  and  (46),  putting  f  for  the  angle 

at  the  star, 

cos  S'.  da  =       da  coa  d  cos  f  -\-  dd  sin  f  ) 

dd'=  —  da  cos  d  sin  f  -f-  dd  cos  f  j 

in  which 

sin  0  sin  (a  +  z  -f  #)  _  sin  0  sin  (a'  —  2  -f-  #') 


sm  Y  = 


cos  $f  cos  d 

cos  0  —  sin  d  sin  d' 

COS  d  COS  3f 


622  PROPER    MOTION    OF    THE   FIXED    STARS. 

Ill  computing  these,  it  will  usually  suffice  to  find  f  by  its  sine 
alone,  since  cos  7*  will  always  be  positive  except  in  the  rare  case 
where  the  star  is  so  near  the  pole  that  cos  0  <  sin  d  sin  d'. 

The  formulae  (665)  are  equally  applicable  whether  da,  dd,  da', 
ddf  denote  the  annual  proper  motion  or  the  whole  proper  motion 
in  the  given  interval. 

EXAMPLE. — The  mean  place  of  Polaris  for  1755  was 
a  ==  10°  55'  44".955  d  =  87°  59'  41".12 

and,  by  the  application  of  the  precession,  this  place  reduced  to 
1820  was  found,  on  page  616,  to  be 

(a')  =  14°  13'  56".85  (£')  =  88°  20'  54".ll 

•        0 

But  the  mean  place  for  1820,  derived  from  observation,  was, 
according  to  BESSEL  in  the  Tabulce.  Regiomontance, 

a'  =  14°  15'  22".575  df  =  88°  20'  54".27 

Hence,  the  proper  motion  from  1755  to  1820,  referred  to  the 
mean  equinox  of  1820,  was 

Aa'  =  -f  85".725  Ar5'  =  -f  0".16 

or  the  annual  motion 

da'  =  -f  1".31885  dd'  =  +  0".00246 

INTow,  to  reduce  this  proper  motion  to  the  year  1755,  we  may 
employ  the  formulae  (665),  by  exchanging  da  with  da'  and  dd 
with  dd',  and  taking  f  with  the  negative  sign,  since  0  is  negative 
for  the  interval  from  1820  to  1755 ;  or  we  may  avpid  the  change 
of  notation  and  of  sign  by  deducing  from  (665)  the  following : 

cos  d .  da  =  da  cos  d'  cos  f  —  (U'  sin  Y 
dd  =  da  cos  d'  sin  f  -f-  dd'  cos  Y 

From  the  example  on  page  616,  we  find 

log  sin  0  =  7.800851  a  +  z  +  0  =  11°  20'  41".57 

with  which  and  d'  =  88°  20'  54".27  we  find 

log  sin  Y  =  8.634966  log  cos  r  =  9.999596 


PROPER    MOTION    OF    THE    FIXED    STARS.  623 

and  hence  for  1755  we  find 

da  =  +  1".08228  dd=  +  0".004098 

If,  now,  there   had   been   given  both  the  mean  place  and  the 
proper  motion  for  1755,  namely, 

a  =  10°  55'  44".955  d  =  87°  59'  41".12 

rfa  =  =  -f  1".08228  dd  =  +  0".004098 

to  find  the  mean  place  for  1820,  we  should  first  take 

(a)  =  10°  55'  44".955  +  1".08228    X  65  =  10°  56'  55".30 
(3)  =  87    59  41  .12    +  0  .004098  X  65  =  87    59  41  .39 

and  employing  these  values,  instead  of  a  and  d,  in  (659)  and  (660), 
we  should  find 

a  _f-  z  +  #  =  A  =  11°  21'  51".92 

log  p  =  9.256691 
A'  —  A  =  2°  28'  33".45 
£(<$'— (S)  =       10'36".44 
whence 

a'  ==  14°  15'  22".57  d'  =  88°  20'  54".27 

as  given  above. 

380.  The  proper  motion  on  a  great  circle. — If  we  denote  this  by 
/>,  and  the  angle  which  the  great  circle  makes  with  the  circle  of 
declination  of  the  star  by  £,  reckoning  the  angle  from  the  north 
towards  the  east,  we  have 

P  sin  %  =  Aa  cos  5  p  cos  /  =  A<? 

Thus,  we  find,  in  the  preceding  example,  for  Polaris  in  1755, 

p  =  0".03809  %  =  83°  49'.4 

and  in  1820, 

p  =  0".03809  *'=86°  17'.8 

where  the  accent  of  ^'  is  used  for  the  second  epoch,  but  p  is 
necessarily  the  same  for  both  epochs. 

It  is  evident,  moreover,  that  we  have  %'  =  %  -j-  p,  and  hence, 
if  p  and  y  have  been  found  for  one  epoch,  it  is  only  necessary  to 
compute  Y  to  obtain  the  reduction  to  another  epoch;  for  we  then 
have,  by  (665), 

cos  df  da  =  p  sin  (/  -j-  y)  =  P  8m  / 
dd'  =  p  cos  (/  -j-  f)  =  p  cos/ 


624  NUTATION. 


NUTATION. 

381.  By  the  luni-solar  precession,  combined  with  the  diminu- 
tion of  the  obliquity  of  the  ecliptic,  the  mean  pole  of  the  equator 
is  carried  around  the  pole  of  the  ecliptic,  but  gradually  approach- 
ing it.  But  the  true  pole  of  the  equator  has  at  the  same  time  a 
small  subordinate  motion  around  the  mean  pole,  which  is  called 
nutation.  This  motion,  if  it  existed  alone,  would  be  nearly  in  an 
ellipse  whose  major  axis  would  be  18". 5  and  minor  axis  13". 7, 
the  major  axis  being  directed  towards  the  pole  of  the  ecliptic ; 
and  a.  revolution  of  the  true  around  the  mean  pole  would  be 
completed  in  a  period  of  about  nineteen  years.  This  period  is 
the  time  of  a  complete  revolution  of  the  moon's  ascending  node 
on  the  ecliptic,  upon  the  position  of  which  the  principal  terms 
of  the  nutation  depend. 

This  periodic  nutation  of  the  pole  involves  a  corresponding 
nutation  of  the  obliquity  of  the  ecliptic  =  AS,  and  a  nutation  of  the 
equinox  in  longitude,  or,  briefly,  a  nutation  in  longitude  =  A^,  which 
are  expressed  by  the  following  formulae*  for  the  year  1800: 

At  =        9".2231  cos  &— 0".0897  cos  2  ^  -f  0".0886  cos  2  <£ 

-fO".5510cos20  +  0".0093co8(O  +  /*) 

(666) 

AX  =  —  17".2405  sin  £+0".2073sin2  &  —  0".2041  sin  2<£  +  0".0677  sin  (£  —  r  ) 
—  1^2694sin2Q+0'\1279sin(0— /*)— 0".0213sin(OH--r) 

in  which 

&  =  the  longitude  of  the  ascending  node  of  the  moon's  orbit, 

referred  to  the  mean  equinox, 
([  =  the  moon's  true  longitude, 
O  =  the  sun's  true  longitude, 
F  =  the  true  longitude  of  the  sun's  perigee, 
Ff  =  the  true  longitude  of  the  moon's  perigee. 

The  quantity  AA  is  also  called  the  equation  of  the  equinoxes. 

The  coefficient  of  cos  &  in  the  formulse  for  AS  is  called  the 
constant  of  nutation.  The  coefficient  of  sin  &  in  the  formula  for 
AA  is  equal  to  this  constant  multiplied  by—  2  cot  2s0,  in  which 

*  PETERS,  Numerus  Constant  Nutationis,  p.  46.  Some  exceedingly  small  terms,  and 
others  of  short  period,  are  here  omitted,  as,  even  if  they  are  not  altogether  insensible 
in  a  single  observation,  their  effect  disappears  in  the  mean  of  a  number  of  observa- 
tions. 


NUTATION.  625 

s0  =  23°  27'  54".2.    These  coefficients,  however,  are  not  absolutely 
constant  :  so  that,  according  to  PETERS,  the  formulae  for  1900  will  b* 

Ae  =        9".2240  cos  £  —  0".0896  cos  2&  +  0".0885  cos  2  £ 

-f  0".  5507  cos2O  +  Q"-0092  cos  (Q  +  -O 

(667) 

AA  —  —  17".2577sin  &  -f  0".  2073  sin  2^—0".  2041  sin  2  <£  -f  0".  0677  sin  (<£  —  r') 
—  1".2695  sin  2  O  +  0".1275  sin  (Q—  f)  —  0". 


Since  the  attractions  of  the  sun  and  moon  upon  the  earth  do  not 
disturb  the  position  of  the  ecliptic,  but  only  that  of  the  equator 
and  its  intersection  with  the  ecliptic,  the  nutation  does  not  affect 
the  latitudes  of  stars,  and  its  effect  upon  their  longitudes  is 
simply  to  increase  them  all  by  the  same  quantity  A^. 

382.  To  find  the  nutation  in  right  ascension  and  declination  for  a 
given  star  at  a  given  time.  —  Let  a  and  3  denote  the  mean  right 
ascension  and  declination  of  the  star  at  the  given  time  ;  a'  and  df 
the  true  right  ascension  and  declination  at  this  time,  or  the  mean 
place  corrected  for  the  nutation.  Let  the  coefficients  of  the 
formulae  for  AS  and  A^  be  found  for  the  given  year  by  interpola- 
tion between  the  values  for  1800  arid  1900,  and  then,  taking 
&  ,  C  ,  O  ,  r,  and  F'  from  the  Ephemeris  for  the  given  date  (the 
day  of  the  year,  and,  for  the  greatest  precision,  the  hour  of  the 
day),  we  can  compute  the  values  of  AS  and  A^.  "We  can  then 
have  either  a  rigorous  or  an  approximate  solution  of  our  problem. 

A  rigorous  solution  may  be  obtained  by  employing  the  for- 
mulae (656),  (658),  and  (659),  substituting  s  -f  J  AS,  AS,  A;.,  a  -f  2, 
and  a'  —  z'  for  \  (s/  -f  st),  s/  —  sl9  ty  —  ^,A  and  A  ',  respectively. 

Another  rigorous  solution  is  obtained  by  first  computing  the 
mean  longitude  X  and  latitude  /?,  from  the  given  a  and  d,  and 
the  mean  obliquity  s,  by  Art.  23.  Then,  with  the  true  longitude 
A  -f-  A^,  the  true  latitude  ft,  and  the  true  obliquity  s  -f  AS,  we  can 
compute  the  true  right  ascension  a/  and  declination  df  by  Art.  26. 

But,  since  AS  and  A^  are  very  small,  an  approximate  solution 
by  means  of  differential  formulEe  will  be  sufficiently  accurate, 
and  practically  more  convenient.  The  effect  of  varying  A  and 
s  by  A^  and  AS,  while  ft  is  constant,  is,  by  the  equations  (53), 

,  COS  V)  COS  ft 

a  —  a  =  AA  .  —   —  --  A£  COS  a  tan  d 
COS  d 

df  —  d  =  AA  .  sin  i)  cos  p  -f-  AS  sin  a 
in  which  y  is  the  position  angle  at  the  star,  as  defined  in  Art.  25. 

VOL.  L—  40 


626  NUTATION. 

Substituting  the  values  of  cos  ^  cos  /?  and  sin  jy  cos  /9  there  given, 
we  have 

a'  —  a  =  AA  (cos  e  -}-  sin  e  sin  a  tan  <5)  —  AS  cos  a  tan  d 
6'  —  d  =  AA  sin  e  cos  a  -j-  As  sin  a 

Hence,  substituting  the  values  of  A^  and  AS  for  1800,  with 
€  =  23°  27'  54",  and  also  the  values  for  1900  with  e  =  23°  27'  7", 
we  find 


—  (15".8148  -j-  6".8650  sin  a  tan  6)  sin  &    —  9".2231  cos  a  tan  6  cos  & 

15".8321  6".8682  9".2240 

-f-  (  0".1902  -f-  0".0825  sin  a  tan  6}  sin  2&  -f  0".0897  cos  a  tan  d  cos  2& 

-  (  0".1872  4-  0".0813  sin  a  tan  6)  sin  2^  —  0".0886  cos  a  tan  6  cos  2<£ 

+  (  0".0621  -j-  0".0270  sin  a  tan  ri)  sin  (C  —  ,T') 

—  (  1".1644  4-  0".5055  sin  a  tan  J)  sin  2Q  —  0".5510  cos  a  tan  6  cos  2Q 
4-  (  0".1173  4-  0".0509  sin  a  tan  J)  sin  (Q  —  r) 

—  (  0".0195  4-  0".0085  sin  a  tan  6)  sin  (Q  4.  r)  —  0".0093  cos  a  tan  6  cos  (Q  4~  -T) 

(668) 
d'  —  <J  =  —  6". 8650  cos  a  sin  &      4~  9". 2231  sin  a  cos  ££ 

6".8682  9".2240 

4-  0".0825  cos  a  sin  2&  —  0".0897  sin  a  cos  2  & 

—  0".0813  cos  a  sin  2<£  4-  0".0886  sin  a  cos  2<£ 
4-  0".0270  cos  a  sin  (C  —  JT') 

—  0".5055  cos  a  sin  2  O  4-  0".5510  sin  a  cos  2  Q 
4-  0".0509  cos  a  sin  (Q  —  r) 

-  0".0085  cos  a  sin  (Q  4-  r)  4-  0".0093  sin  a  cos  (Q  4-  r) 

The  values  of  the  coefficients  which  sensibly  change  during  the 
century  are  given  for  1900  in  small  figures  below  the  values  for 
1800.* 

Previous  to  the  investigations  of  PETERS,  the  only  terms 
retained  in  the  nutation  formula  were  those  .depending  on 
& ,  2  & ,  2  <c ,  and  2  O .  Of  the  additional  terms  added  by  him,  I 
have  retained  only  those  which  can  have  any  sensible  effect  in 
the  actual  state  of  the  art  of  astronomical  observation. 

383.  General  tables  for  the  nutation  in  right  ascension  and  declina- 
tion.— Of  the  various  tables  proposed  for  facilitating  the  compu- 

*  If  we  take  into  account  the  squares  of  AA  and  A£  and  their  product  in  the  develop- 
ment of  a' —  a  and  6'  —  6  in  series,  some  of  the  coefficients  are  changed,  but  only  by 
two  or  three  units  in  the  last  decimal  place.  Compare  the  formulas  of  the  text  with 
those  given  by  PETERS  in  the  Numerus  Constans.  and  by  STRUVE  in  the  Astronom. 
Nach.,  No.  486. 


NUTATION.  627 

tation  of  the  nutation  formulae,  the  most  compendious  are  those 
computed  by  NICOLAI,  according  to  the  form  suggested  by  GAUSS, 
and  included  in  WARNSTORFF'S  edition  of  SCHUMACHER'S  Hiilfs- 
tafeln.  In  these  tables  the  new  constants  are  adopted  from 
PETERS,  as  in  the  preceding  formulae,  and  the  epoch  is  1850. 

For  the  lunar  nutation  in  right  ascension,  the  first  table  gives, 
with  the  argument  ft ,  the  quantity 

—  15".8235  sin  ft  =  'c 

The  two  remaining  terms  in  the  first  line  of  our  formula  are 
reduced  to  a  single  term  by  assuming  auxiliaries  b  and  B,  also 
given  in  the  tables  with  the  argument  ft ,  determined  by  the 

conditions 

b  sin  (ft  +  £)  =  6".8666  sin  ft 
b  cos  (ft  +  J5)  =  9".2235  cos  ft 

Thus,  the  first  line  of  the  formula,  containing  the  principal  terms 
of  the  lunar  nutation  in  right  ascension,  becomes 

c  —  b  cos  (ft  -}-  B  —  a)  tan  d 

By  the  use  of  the  same  auxiliaries,  the  first  two  terms  of  the 
lunar  nutation  in  declination  are  reduced  to  the  following : 

-  b  sin  (ft  -f  B  —  a) 

For  the  solar  nutation,  the  second  table  gives,  with  the  argu- 
ment 2Q,  the  quantity 

-1".1644  sin  20=0 

and  the  two  remaining  terms  involving  2O  are  reduced  to  a 
single  one  by  the  auxiliaries /and  F,  given  in  the  table,  which 
are  determined  by  the  conditions 

/  sin  (20  +  F)  =  0".5055  sin  2Q 
/  cos  (2  O  +  F)  =  0".5510  cos  2  0 

so  that  the  solar  nutation  in  right  ascension  is 
g  —/cos  (20  -f-  F  —  a)  tan  d 
and  the  solar  nutation  in  declination  is 

-/sin  (20  -f  F—  a) 

Almost  all  the  remaining  terms  of  the  formulse  may  also  be 
found  by  means  of  the  table  for  the  solar  nutation.  The  coeffi- 
cients of  the  terms  in  2  ft  and  2  £  are  about  one-sixth  part  of 


628  ABERRATION. 

those  of  the  terms  in  20,  while  the  signs  of  the  terms  in  2& 
are  the  opposite  to  those  in  20  :  hence,  to  find  the  value  of 
these  terms,  we  can  enter  the  table  first  with  the  argument 
2&+  180°  (=  2&-f  VI*),  and  then  with  2C;  and,  computing  the 
nutation  in  each  case  by  the  above  forms  for  the  solar  nutation, 
take  J,  or  more  exactly  ^67,  of  the  sum  of  the  results.  The  terms 
in  O  -f  P  are  obtained  by  entering  the  table  with  the  argument 
O  +  P  and  taking  ^  of  the  results.  The  terms  in  O  — .Twill 
be  found  in  the  most  simple  manner  by  multiplying  the  annual 
precession  [given  by  (663),  and  usually  computed  in  connection 
with  the  nutation]  by  ^  sin  ( O  —  P) ;  and  the  terms  in  £  —  P' 
by  multiplying  the  annual  precession  by  y^  sin  ( <C  —  71'). 

The  computation  even  with  the  aid  of  these  tables  is  suffi- 
ciently tedious.  Their  chief  recommendation  is  their  brevity; 
but  where  the  nutation  is  to  be  computed  very  frequently,  more 
extended  tables  are  required,  such,  for  example,  as  are  given 
in  the  3d  vol.  of  the  Washington  Observations,  Appendix  C,  by 
Professors  HUBBARD,  COFFIN,  and  KEITH. 

ABERRATION. 

384.  The  apparent  direction  of  a  star  from  the  earth  is  deter- 
mined by  the  direction  of  the  telescope  through  which  it  is  ob- 
served. But  this  apparent  direction  differs  from  the  true  one  in 
consequence  of  the  motion  of  the  earth  combined  with  the  pro- 
gressive motion  of  light;  for  the  telescope,  partaking  of  the 
movement  of  the  earth,  is  changing  its  position  while  the  light 
is  descending  through  its  axis. 

Let  us  distinguish  between  the  two  instants  t  and  tf  when  the 

O 

ray  of  light  from  the  star  arrives  respectively  at  the  object-end 
and  at  the  eye-end  of  the  axis  of*the  telescope. 
Let  A,  B  (Fig.  57)  be  the  position  of  the  object 
and  eye  end  of  the  telescope  at  the  instant  t; 
Af,  Bf,  their  positions  at  the  instant  t1 ;  BB1 ',  the 
motion  of  the  earth  in  the  interval  t'  —  t,  in 
which  the  ray  SAB'  from  the  star  is  describing 
the  line  AB'.  Then  it  is  evident  that,  while  B'A 
is  the  true  direction  of  the  star,  B'A'  is  the  ap- 
parent direction  as  given  by  the  telescope.* 
'  Moreover,  supposing  the  motion  of  the  earth  for 


B    b    B 


*  GAUSS  :    Theoria  Motus  Corporum  Coelestium,  p. 


ABERRATION.  629 

so  small  an  interval  to  be  rectilinear  and  uniform,  and  the 
motion  of  light  to  be  uniform,  the  lines  BA  and  B'A'  are 
parallel,  and  the  ray  of  light  during  its  progress  from  A  to  B' ', 
is  constantly  in  the  axis  of  the  telescope ;  for  instance,  when  the 
telescope  is  in  the  position  ba,  the  ray  will  have  reached  the 

point  a,  and  we  have 

AaiBb  =  AB'\BBr 

The  difference  of  apparent  direction  thus  caused  by  the 
motion  of  the  earth  combined  with  that  of  light  is  called  the 
aberration  of  the  fixed  stars.  When  we  also  take  into  account  the 
motion  of  the  luminous  body,  as  in  the  case  of  the  planets, 
another  species  of  aberration  occurs,  which  will  be  considered 
hereafter,  under  the  name  of  the  planetary  aberration. 

The  whole  displacement  of  the  star  produced  by  aberration  is 
in  the  plane  passed  through  the  star  and  the»line  of  the  observer's 
motion,  and  the  star  appears  to  be  thrown  forward  in  this  plane 
in  the  direction  of  that  motion.  Thus,  in  the  figure  the  whole 
aberration  is  the  angle  SB' A' ;  and,  if  we  conceive  the  plane  of 
the  lines  SB'  and  BB'  to  be  produced  to  the  celestial  sphere, 
this  plane  will  be  that  of  a  great  circle  drawn  through  the  place 
of  the  star  and  the  points  of  the  sphere  in  which  the  line  BBf 
meets  it.  The  displacement  of  the  star  will  be  the  arc  of  this 
circle  subtending  the  angle  SB' A'  and  measured  from  the  star 
towards  that  point  of  the  sphere  towards  which  the  observer  is 
moving. 

385.  To  find  the  aberration  of  a  star  in  the  direction  of  the  observer's 
motion. — Let 

#  =  AB'Bl  =  the  true  direction  of  the  star  referred  to  the 

line  B'BV 
=  the  arc  of  a  great  circle  of  the  sphere  joining  the  star's 

true  place  and  the  point  from  which  the  observer  is 

moving, 
#'  =  the  apparent  direction  of  the  star  referred  to  the  same 

line,  ==  ABBV 
Y=  the  velocity  of  light, 
v  =  the  velocity  of  the  observer; 

then  the  aberration  in  the  plane  of  motion  is  the  angle  A'B'A 
=  B'AB  =  #'  —  #,  and  the  triangle  ABB'  gives 
Bin  (#'  _  ,9)  __  BB'  __  v_ 
sin  #         ~~AB'~~  V 


630  ABERRATION. 

As  $'  — -  $  is  very  small,  we  may  put  the  arc  for  the  sine  :  and  if 
we  then  also  put 

*  =  F£P  (669> 

we  shall  have 

#  _  #  —  A-  sin  #'  (670) 

where  the  constant  k  may  he  regarded  as  known  from  the  velo- 
cities of  light  and  of  the  observer. 

386.  The  motion  of  the  observer  on  the  surface  of  the  earth  is 
the  resultant  of  the  motion  of  the  earth  in  its  orbit  and  its  rota- 
tion on  its  axis ;  that  is,  of  its  annual  and  diurnal  motions.    These 
may  be  separately  considered. 

The  annual  aberratibn  is  that  part  of  the  total  aberration  which 
results  from  the  earth's  annual  motion.  It  may  be  called  the 
aberration  for  the  earth's  centre. 

The  diurnal  aberration  is  that  part  of  the  total  aberration  which 
results  from  the  earth's  diurnal  motion.  It  obviously  varies 
with  the  position  of  the  observer  011  the  earth's  surface,  and 
vanishes  for  an  observer  at  the  poles. 

387.  To  faid  the  annual  aberration  of  a  star  in  longitude  and  lati- 
tude.— Let 

lf  fi  =  the  true  longitude  and  latitude  of  the  star, 

X' ',  /?'  =  the    apparent   longitude    and   latitude    (affected    by 

aberration), 
O  =  the  true  longitude  of  the  sun. 

The  point  of  the  sphere  from  which  the  earth  appears  to  be 
moving  is  a  point  in  the  ecliptic  whose  longitude  is  90°  +  O 
(the  eccentricity  of  the  earth's  orbit  being  here  neglected),  and 

the  mean  velocity  of  the  earth  in  its  orbit 

may  be  supposed  to  be  substituted  in  (669) : 

so  that  k  is  known. 

If,  then,  BE  (Fig.  58)    is    an  arc  of  the 

ecliptic,  jE'the  point  from  which  the  earth 
is  moving,  S  the  true  place  of  the  star,  and  if  SB  is  drawn  per- 
pendicular to  BE,  we  have,  in  the  right  triangle  SBE, 

SB  =  ?,  BE  =  9a°  -f  O  —  *, 


ABERRATION.  631 

and  hence,  if  we  denote  the  angle  E  by  f,  we  have 

sin  #  sin  f  =       sin  ft  "| 

sin  &  cos  Y  =       cos  ft  cos  (O  —  A)  >    (671) 

cos  #  =  —  cos  ft  sin  (O  —  ^)  J 

The  apparent  place  of  the  star  is  on  the  great  circle  JES  at  the 
distance  #'  from  S:  so  that,  if  we  now  suppose  S  to  be  the 
apparent  place,  the  angle  f  is  not  changed,  and  we  have 

sin  $'sin  Y  =       sin  ft'  ^ 

sin  #'  cos  Y  =       cos  /9'  cos  (O  —  A')  V    (672) 

cos  #'  =  —  cos  /3'  sin  (O  —  >*')  J 

If,  then,  the  true  place  of  the  star  is  given,  the  equations  (671) 
may  be  used  to  determine  f  and  & ;  then  $'  will  be  found  from 
(670),  and,  finally,  X'  and  ft'  will  be  found  from  (672).  This  is 
the  direct  and  rigorous  solution  of  the  problem ;  but  a  more 
convenient  solution  is  obtained  by  eliminating  &  and  f  as  follows. 
We  find,  from  the  equations  (671)  and  (672), 

sin  $  cos  »9  cos  y  =  —  cos  ft  cos  ft'  sin  (O  —  A)  cos  (O  —  ^') 
sin  &  cos  #'  cos  Y  =  —  cos  ft  cos  ft'  cos  (O  —  ^)  sin  (O  —  A') 

the  difference  of  which  is 

sin  (  #'  —  #)  cos  ^  =  —  cos  ft  cos  /9'  sin  (/  —  A) 
whence 

(#'  —  #)  cos  ^  _         k  sin  #'  cos  Y 

cos  /3  cos  /5'  cos  ft  cos  /5r 

or 

(673) 


COS  , 

Again,  we  find,  from  our  equations, 

cot  r  —  cot  ftf  cos  (O  —  A')  —  cot  /?  cos  (0  —  A) 

by  which  ft'  can  be  found  from  /?  after  X  has  been  found  by  (673), 
or  we  may  find  the  difference  between  ft1  and  ft  thus  : 


2  sin  |  (Ay  —  A)  sin  [Q  —  j  (/+  A)]  sin  /3'  cos  £ 
105    ~^=  cos(O-A') 

whence,  taking  2  sin  J(A'  —  ^)  =  sm  (^'  ~~  ^)»  we  obtain,  by  means 
of  (673), 

ftf  —  ft  =  —  A  sin  [O  —  J  (*'  "f  A)l  sin  /5' 


632  ABERRATION. 

The  equations  (673)  and  (674)  are  almost  rigorously  exact  ;  but, 
since  the  value  of  A'  is  only  about  20",  a  sufficient  degree  of 
accuracy  will  be  obtained  if  in  the  second  members  we  put 
^  and  ft  for  ^  and  /9'.  The  formulae  for  the  annual  aberration  in 
longitude  and  latitude  thus  become 

A'  -  A  =  -  k  cos  (Q  -  A)  sec  0  \    • 

0'  —  /5  =  —  k  sin  (O  —  *)  sin  £  J 

in  which  the  value  of  the  constant,  according  to  STRUVE,*  is 

k  =  20".4451 

These  last  formulae  may  be  directly  deduced  by  differentiating 
the  equations  (671). 

If  we  retain  terms  of  the  second  order  in  developing  (673) 
and  (674),  \ve  shall  find  that  the  following  quantities  will  be 
added  to  the  second  members  of  (675)  : 


—  I  A2  sin  1"  sin  2  (Q  —  A) 
and  -  ±  A-2  sin  V  tan  0  —  1  A2  sin  1"  cos  2  (0  —  A)  tan  /? 

But  the  term  —  J  A2sin  l"tan/9  being  constant  may  be  omitted, 
since  it  will  be  included  in  the  expression  of  the  star's  mean 
place,  which  (Art.  361)  involves  the  non-periodic  elements  of 
the  star's  position.  Retaining,  therefore,  only  the  periodic  terms 
—  namely,  those  involving  O  —  the  more  complete  formulae  will  be 

A'  —  -A  =  —  20".4451  cos  (O  —  *)  sec  /3  —  0".0010133  sin  2  (0  —  /t)  sec2  /?  >  (675*\ 
/?'—  /3=  —  20".4451  sin  (Q  —  *)  sin  J3  —  0".00050G7  cos  2  (Q  —  A)  tan  0  /  * 

The  last  terms  will  be  sensible  only  for  stars  very  near  the  pole. 
Terms  of  the  second  order  not  multiplied  by  tan  /?  or  sec  /9  are 
wholly  insensible,  and  have  been  disregarded  in  the  deduction 
of  the  above  formulae. 

388.  It  is  easy  to  prove,  from  the  equations  (675),  that  the 
effect  of  the  aberration  is  the  same  as  if  the  star  actually  moved 
in  a  circle  parallel  to  the  plane  of  the  ecliptic  ;  the  diameter  of 
the  circle  being  equal  to  the  distance  of  the  star  multiplied  by 
;«in  k.  This  circle  will  be  seen  projected  upon  a  plane  tangent 
to  the  sphere  at  the  mean  place  of  the  star,  as  an  ellipse  whose 
:major  axis  is  sin  A;  and  minor  axis  sin  k  sin  ft,  the  radius  of  the 

*  Astron.  Nach.,  No.  484. 


ABERRATION.  633 

sphere  being  unity.  The  period  in  which  a  star  appears  to 
describe  this  ellipse  is  a  sidereal  year. 

389.  To  find  the  annual  aberration  in  right  ascension  and  declina- 
tion.— Let 

A,  D  =  the  right  ascension  and  declination  of  the  point  E 
(from  which  the  earth  is  moving) ; 

then,  in  the  triangle  formed  by  the  point  jEJ,  the  star,  and  the 
pole  of  the  equator,  the  sides  are  90°  —  D,  90°  —  d,  and  & ;  and 
the  angle  opposite  to  $  is  A  —  a.  If  then  we  suppose  the  side 
$  to  vary,  the  corresponding  variations  of  the  angle  A  —  a  and 
the  side  90°  —  d  may  be  directly  deduced  by  the  differential 
formulae  of  Art.  34.  The  angle  at  E  and  the  side  90°  —  D  being 
constant,  we  find 

cos  d  .  da  =  —  d$  sin  C 
dd  =  —  d&  cos  G 

where  C  denotes  the  angle  at  the  star.  For  determining  (7,  our 
triangle  gives 

sin  #  sin  C  =  cosD  sin  (A  —  a) 

sin  &  cos  C  =  cos  d  sinD  —  sin  d  cos  D  cos  (A  —  a) 

In  (670)  we  may  employ  sin  $  for  sin  &' :  so  that,  putting  a'  —  a 
and  $'  —  &  for  da  and  d&,  we  find 

a!  —  a  =  —  k  sec  d  cosD  sin  (A  —  a)  1 

8'  —  d  =  —  k  [cos  dsinD  —  sin  d  cosD  cos  (A  —  a)]  / 

The  quantities  A  and  D  are  found  from  the  right  triangle 
formed  by  the  equator,  the  ecliptic,  and  the  declination  circle 
drawn  through  E,  by  the  formulae, 

cosDcosJ.^  —  sin  O  "} 

cosD  sin  A  =       cos  O  cos  e  >    (677) 

sin  D  =       cos  O  sin  e  J 

If  we  substitute  these  values  in  the  formulae  for  a'  —  a  and 
df  —  d,  after  developing  sin  (A  —  a)  and  cos  (A  —  a),  we  obtain 

a!  —  a  =  —  k  sec  d  (cos  O  cos  e  cos  a  -}-  sin  O  sin  a)  "| 
d'  —  d  =  —  k  cos  O  (sin  e  cos  3  —  cos  e  sin  d  sin  a)      >    (678) 
—  A  sin  O  sin  d  cos  a  J 


634  ABERRATION. 

If  we  retain  the  terms  of  the  second  order,  (omitting,  however, 
those  which  do  not  involve  0,  or  the  non-periodic  terms),  we 
find  that  the  aberration  in  right  ascension  obtains  the  additional 
terms 

-  \  &2  sin  1"  (1  -f-  cos2  e)  cos  2  0  sin  2  a  sec2  d 
-f-  \  A'2  sin  1"  cos  e  sin  2  O  cos  2  a  sec2  d 

and  the  aberration  in  declination  the  terms 

*  -f  |  A2  sin  V  [sin2  e  —  (1  -f  cos2  e)  cos  2  O  cos  2  a]  tan  <5 
—  I  A2  sin  1"  cos  e  sin  2  O  sin  2  a  tan  d 

Substituting  the  value  of  k  in  these  terms,  together  with 
e  =  23°  27'  30"  (for  1850),  and  omitting  insensible  quantities, 
the  corrections  of  the  formulae  (678)  will  be 

in  (af  —  a),         —  0".000931  sin  2  (O  —  a)  sec2  d 
in  (<r  _  3),          -  0".000466  cos  2  (O  —  a)  tan  d 

EXAMPLE  1. — The  mean  longitude  and  latitude  of  Spica  for 
January  10,  1860,  are 

A  =  201°  53'  22".33  ft  =  —  2°  2'  36".29 

and  the  sun's  longitude  is 

0  =  289°  30' 

Hence,  we  find,  by  (675),  the  aberration  in  longitude  and  latitude, 

X  —  A  =  —  0".85  /?'  —  ft  =  +  0".73 

The  corresponding  mean  right  ascension  and  declination  are 

a  =  13»  17"  49'.62  d  =  —  10°  25'  44".9 

whence,  by  (678),  taking  e  =  23°  27'.4,  we  find  the  aberration  in 
right  ascension  and  declination, 

a'  —  a  =±  —  0".53  ==  —  0'.035  d'  —  (5  =  +  0".99 

EXAMPLE  2. — The  mean  place  of  Polaris  for  1820.0  was 

a  =  0»  57m  1-.505  =  14°  15'  22".57 
d  =  88°  20'  54".27 

and  for  this  date, 

Q=280°0'  e  =  23°  27'.8 


ABERRATION.  G35 

with  which  the  aberration  in  right  ascension  and  decimation  is 
found,  by  (678),  to  be? 

0'  _  a  =  4-  62".51  =  +  4M67  3'  —  8  =  +  20".27 

The    additional    terms    of  (678*)  are  in  this   case  —  0".158  = 
-  O'.Oll  and  -+-  0".016,  and  the  more  correct  values  are,  there- 
fore, 

a'  —  a  =  -f  4M56  3'— 3=  +  20".29 


390.  Gauss's  Tables  for  computing  the  aberration  in  right  ascension 
and  declination. — If  we  determine  a  and  A  by  the  conditions 

a  sin  (O  +  ^)  =  ^  s^n  O 

a  cos  (O  +  A)  =  k  cos  O  cos  e 

the  formulae  (678)  may  be  expressed  as  follows : 

a'  —  a  =  —  a  sec  3  cos(O  +  A  —  a) 

3'  —  8  =  —  a  sin  3  sin  (Q  -f  A  —  a)  —  k  cos  Q  cos  §  sin  e 

=  —  a  sin  d  sin  (O  +  A  —  a)  —  £  k  sin  s  cos  (O  +  <5) 

—  £  /:  sin  e  cos  (O  —  <0 

The  first  of  the  tables  proposed  by  GAUSS*  gives  A  and  log  a 
with  the  argument  sun's  longitude,  and  with  these  quantities  we 
readily  compute  the  aberration  in  right  ascension  and  the  first 
part  of  the  aberration  in  declination.  The  second  and  third 
parts  of  the  aberration  in  declination  are  taken  directly  from 
the  second  table  with  the  arguments  O  -h  #  and  O  —  d.  The 
tables  have  been  recomputed  by  NICOLAI  with  the  constant 
k  =  20". 4451,  and  are  given  in  WARNSTORFF'S  edition  of  SCHU- 
MACHER'S Hulfstafeln. 

The  value  of  e  for  1850  is  employed  in  computing  these 
tables.  The  rate  of  change  of  e  is  so  slow  that  the  tables  will 
answer  for  the  whole  of  the  present  century,  unless  more  than 
usual  precision  is  desired. 

391.  In  the  preceding  investigation  of  the  aberration  formulae 
we  have,  for  greater  simplicity,  assumed  the  earth's  orbit  to  be  a 
circle  and  its  motion  in  the  orbit  uniform.     Let  us  now  inquire 
what  correction  these  formulae  will  require  when  the  true  ellip- 
tical motion  is  employed. 

*  Monatliche  Corresponded,  XVII.  p.  312. 


636  ABERRATION. 

If  it  is  the  true  anomaly  of  the  earth  in  the  orbit,  reckoned 
from  the  perihelion,  at  the  time  t  from  the  perihelion  passage, 
r  the  radius  vector,  a  the  mean  distance  of  the  earth  from  the 
sun,  or  the  semi-major  axis  of  the  ellipse,  we  have 

a  (1  —  e^ 


1  -j-  e  cos  u 

The  true  direction  of  the  earth's  motion  at  any  time  is  not,  as 
in  the  circular  orbit,  at  right  angles  to  the  direction  of  the  sun, 
but  in  that  of  the  tangent  to  the  curve.  If  we  denote  the  angle 
which  the  tangent  makes  with  the  radius  vector  by  90°  —  i,  we 
have,  by  the  theory  of  curves, 

cot  (90°  —  t)  =  _•  -- 
J      r  du 

whence,  by  the  above  equation  of  the  ellipse, 

e  sin  u 

tan  i  =  - 

1  -j-  e  cos  u 

and  the  true  direction  of  the  earth's  motion  will  be  taken  into 
account  in  our  formulae  (675),  if  for  O  we  substitute  Q  —  i. 

If  vl  denotes  the  true  velocity  of  the  earth  in  its  orbit  at  the 
time  t,  we  have 

.du 
v1=  r  sec  i  — 

and  if  /  is  the  area  described  by  the  radius  vector  in  the  time  t, 
F  the  whole  area  of  the  ellipse  described  in  the  period  JJ  we 
have,  by  KEPLER'S  first  law, 

/        F 

t  "  T  - 
or 


We  also  have,  by  the  theory  of  the  ellipse, 

F=na?y\l  —  e2) 

df__r*_    du 
dt~~~2"di 
and  hence 

du  _  2?rfl21/(l  —  I2) 


ABERRATION.  637 

which,  substituted  in  the  above  value  of  vv  together  with  the 
value  of  r,  gives 

a— -  —  •(!  +  «  «>s  tOseci 


The  mean  value  of  this  velocity  is  that  value  which  it  would 
have  if  the  small  periodic  terms  depending  on  u  and  i  were 
omitted  (Art.  361)  ;  thus,  denoting  the  mean  velocity  by  v,  we 
have 

v  =  -  -  --  -  (679) 


v1  =  v  (1  -f  e  c°8  u)  sec  *  (680) 

If,  then,  V  is  the  velocity  of  light,  and  we  put 

kj_  =  -  :  --  -  =  k  (1  -f-  e  cos  u)  sec  i 

we  can  at  once  adapt  our  equations  (675)  to  the  case  of  the 
elliptical  orbit,  by  introducing  kv  for  k  and  O  —  i  for  O  ,  so  that 
we  have 

X'  —  X  =  —  #(1  -f-  e  cos  w)  cos  (O  —  X  —  0  8ec  ?  sec  P 
ft'  —  p  =  —  k(l  -}-  e  cos  u}  sin  (0  —  ^  —  0  sec  i  sin  /? 

Developing  the  sine  and  cosine  of  (O  —  K)  —  i,  we  have 

cos  (O  —  A.  —  0  sec  i  =  cos  (O  —  ^)  -+-  sin  (O  —  X)  tan  i 
sin  (o  —  I  —  0  sec  i  =  sin  (0  —  A)  —  cos  (O  —  *)  tan  i 

and  substituting  the  value  of  tan  i,  we  find 

A'  —  /I  =  —  k  cos  (O  —  X)  sec  /9  —  he  cos  (0  —  u  —  A)  sec  /? 
P'—p=z  —  #sin  (O  —  A)  sin  /?  —  A'e  sin  (0  —  w  —  A)  sin  p 

The  longitude  of  the  earth's  perigee  is 


by  the  introduction  of  which  we  have,  finally, 

/  —  X  =  —  k  cos  (O  —  A)  sec  0  —  ke  cos  (/*—  A)  sec  /5  1         gl 
/5'  —  0=  —  ^sin(Q  —  /I)  sin  /?  —  fo  sin  (F—  A)  sin  /9  / 

These   formulae   diiFer   from  (675)  only  by  the   second   terms, 
which  therefore  are  the  corrections  for  the  eccentricity  of  the 


638  ABERRATION. 

orbit.  But  we  observe  that  these  terms  involve  only  quantities 
which  for  a  fixed  star  are  very  nearly  constant,  so  that  for  the 
same  star  they  will  have,  sensibly,  the  same  values  for  very  long 
periods  :  the  corrections  themselves  being  exceedingly  small, 
since  e  =  0.01677,  and  hence  ke  =  0".3429.  They  may,  there- 
fore, be  regarded  either  as  constant  corrections,  or  as  corrections 
having  only  a  slow  secular  change  ;  and  in  either  case  they  will 
be  combined  with  the  mean  place  of  the  star,  and  may  be 
altogether  disregarded  in  the  correction  for  the  annual  aberra- 
tion.* The  formulae  (675),  derived  from  the  circular  orbit,  will 
therefore  be  considered  as  complete  (for  the  fixed  stars),  and, 
consequently,  also  (678),  which  are  derived  from  the  same  hypo- 
thesis. 

392.  The  sun's  aberration.  —  Since  /?  is  less  than  1",  there  is  no 
sensible   aberration   in  latitude.     The  aberration   in   longitude 
must  be  found  by  the  complete  formula  (681),  for  in  the  case  of 
the  sun  ^  is  variable.     Hence,  writing  Q  for  \  the  aberration  of 
the  sun  is  found  by  the  formula 

O'—  0  =  —  20".4451  —  0".3429  cos  (71  —  0)  (682) 

in  which  for  this  century  we  may  employ  F  =  280°  without  an 
error  of  0".01. 

We  could  derive,  from  this,  formulae  for  the  sun's  aberration  in 
right  ascension  and  declination  ;  but  the  practical  method  is  to 
treat  the  sun  as  a  planet,  and  to  employ  the  planetary  aberration 
which  is  given  in  a  subsequent  article. 

393.  To  find  the  diurnal  aberration  in  right  ascension  and  declina- 

tion. —  Let 

* 

v'  =  the  velocity  of  a  point  of  the  terrestrial  equator,  arising 
from  the  rotation  of  the  earth, 

V'  *•-  (683) 


^       „, 

Fsm  1"  v 

The  diurnal  aberration  in  the  places  of  stars,  as  observed  from  a 
point  on  the  equator,  may  be  investigated  in  the  same  manner 
as  the  annual  aberration,  by  substituting  the  equator  for  the 
ecliptic,  and,  consequently,  right  ascensions  and  declinations  for 

*  BESSEL,  Tabulae.  Regiomontanse,  XIX. 


ABERRATION.  639 

latitudes  and  longitudes.  The  nadir  of  the  point  of  observation 
is  then  to  be  substituted  in  the  place  of  the  sun  :*  so  that  if  we 
put 

0  =  the  right  ascension  of  the  zenith,  or  the  sidereal  time, 

the  formulae  (675)  are  rendered  immediately  applicable  to  the 
present  case  by  putting  180°  -(-  0,  a,  £,  and  kf  for  Q>  h  &  and  k; 
whence  we  have,  for  a  point  on  the  terrestrial  equator  , 

a!  —  a  =  kr  COS  (0  —  a)  SCC  d 

f  —  d  =  kf  sin  (0  —  a)  sin  d 

Since  every  point  on  the  surface  of  the  earth  moves  in  a  plane 
parallel  to  the  equator,  and  this  plane  is  to  be  regarded  as  coin- 
cident with  the  plane  of  the  celestial  equator,  the  same  formulae 
are  applicable  to  every  point,  provided  we  introduce  into  the  ex- 
pression ^of  k'  the  actual  velocity  of  the  point.  This  velocity 
varies  directly  with  the  circumference  of  the  parallel  of  latitude, 
or  with  its  radius  ;  and  this  radius  for  the  latitude  <p  is  p  cos  ^>', 
<pr  being  the  geocentric  latitude  and  p  the  radius  of  the  earth  for 
that  latitude.  Hence  we  have  only  to  put  vrp  cos  (p'  for  v',  or 
k'p  cos  <p'  for  A/,  and  we  obtain  for  the  diurnal  aberration  in  right 
ascension  and  declination,  for  any  point  of  the  earth's  surface, 
the  formulae 

a!  —  a  =  k'p  COS  <p'  COS  (0  —  a)  S6C  d  } 

3'—d  =  k'f>  cos  ?'  sin  (0  —  a)  sin  d  j 

It  only  remains  to  determine  kf.     For  this  purpose,  we  have, 
by  (679), 

a  IK 


which,  if  T  is  the  length  of  the  sidereal  year  in  sidereal  days, 
gives  the  value  of  v  for  one  sidereal  day.  The  motion  of  a  point 
on  the  earth's  equator  in  one  sidereal  day  is  equal  to  the  circum- 
ference of  the  equator:  so  that,  if  a'  is  the  equatorial  radius,  we 
have  the  value  of  v'  referred  to  the  same  unit  as  v,  by  the 
formula 


*  For  the  observer  is  moving  directly  from  the  west  point  of  his  horizon,  which  is 
90°  of  right  ascension  in  advance  of  the  nadir  ;  and  the  point  from  which  the  earth 
in  its  annual  revolution  is  moving  is  90°  of  longitude  in  advance  of  the  sun. 


640  ABERRATION. 

whence 

v'        Ta'  1/1  — 


But  if  we  put 

p  =  the  sun's  mean  horizontal  parallax, 
we  have 

and  hence  we  find 


or,  taking  STRUVE'S  value  of  k  =  20".  4451,  BESSEL'S  value  of 
7T  =  '36e*.25637,  ENCKE'S  value  of  p  =  8".57116,  and  the  eccen- 
tricity e  =  0.01677, 

k  =;  0".31112 

This  quantity  is  so  small  that  we  may  in  (684)  employ,  cos  tp  for 
pcostp'  without  sensible  error;  and  hence  the  diurnal  aberration 
may  be  found  by  the  formulae 

a'  —  a  ==  0".311  COS  <p  COS  (0  —  a)  SCC  d  1 

d'—  3  =  0".311  cos  v  sin  (0  —  a)  sin  5  / 

The  quantity  0  —  a  is  the  hour  angle  of  the  star;  whence  it 
follows  that  the  diurnal  aberration  in  right  ascension  for  a  star 
on  the  meridian  is  -f-  0".311  cos  (p  sec  d  =  -f  0'.0207  cos  tp  sec  d  ; 
and  the  diurnal  aberration  in  declination  is  then  zero. 

394.  The  illustration  given  in  Art.  388   applies  also  to  the 
diurnal  aberration.     In  one  sidereal  day  each  star  appears  to 
describe   a  small   ellipse  whose  major  axis   is  sin  kf  cos  ^>,  and 
minor  axis  sin  k'  cos  tp  sin  £,  the  radius  of  the  sphefe  being  unity. 
For  an  observer  at  the  pole,  where  cos  tp  =  0,  this  ellipse  becomes 
a  point,  and  the  diurnal  aberration  disappears. 

395.  The  velocity  of  light.  —  The  constant  k  was  determined  by 
STRUVE  by  a  comparison  of  the  apparent  places  of  stars  at  differ- 
ent seasons  of  the  year,  and  not  from  the  known  velocity  of  light. 
We  can,  therefore,  determine  the  velocity  of  light   from   this 
constant.     We  have,  from  the  preceding  articles, 


Y— 


A-' sin  I"         Tsin^  KH*j/(l  —  e2) 


ABERRATION.  641 

in  which,  if  we  take  v'  —  the  velocity  per  second  of  a  point  of 
the  earth's  equator  resulting  from  the  diurnal  rotation,  Fwill 

be  the  velocity  of  light  per  second.     If,  then,  we  take  vf=  — 

we  have  the  following  formula  for  determining  the  velocity  of 
light  from  the  aberration  constant : 

' 

e2) 

This  will  give  the  velocity  in  one  sidereal  or  one  mean  second, 
according  as  we  take  n  —  86400  or  n  =  86164,  the  number  of 
seconds  of  either  kind  in  a  sidereal  day.  With  BESSEL'S  value 
of  the  equatorial  radius,  Art.  80,  and  the  values  of  T,  p,  A',  and 
6,  above  employed,  we  find 

in  one  sid.  second,      V—  191058  miles  =  307473000  metres; 
in  one  mean  second,  V==  191581  miles  =  308314000  metres. 

The  time  required  by  light  to  traverse  the  mean  distance  of 
the  earth  from  the  sun  is 


a         nT sink -i/I  —  e*       An-  *.Q       0 

—  =  -       £        -  =  497*.78  =  8m  17'.78  mean  time. 

396.  Planetary  aberration. — When  the  observed  body  is  a  planet, 
and,  therefore,  in  motion  relatively  to  the  earth,  the  aberration 
above  considered  is  not  the  complete  aberration ;  but  we  must 
further  take  into  account  the  time  required  by  light  to  come 
from  the  planet  to  the  earth ;  for  in  this  time  the  planet  will 
have  sensibly  changed  its  place.  Let  us 
suppose  that  the  ray  of  light  which  reaches  p  Flg'«\9* 

the  telescope  at  the  time  t  left  the  planet 
at  the  time  T;  let  P  (Fig.  59)  be  the 
planet's  place  in  space  at  the  time  T9  and 
p  its  place  at  the  time  t;  A  the  place  of 
the  object-end  of  the  telescope  at  the  time 
Tj  a  its  place  at  the  time  £,  ab  the  direction 
of  the  axis  of  the  telescope  at  the  time  ty 
afbf  the  position  of  the  axis  at  the  time  V  when  the  light  reaches 
the  eye-end  of  the  telescope.  Then  it  is  evident  that 

1st.  The  straight  line  AP  gives  the  true  direction  of  the  planet 

at  the  time  T; 
VOL.  L— 41 


642  ABERRATION. 

2d.    The  straight  line  ap  gives  the  true  direction  at  the  time  t ; 

3d.  The  straight  line  ba  or  b'a'  gives  the  apparent  direction  at 
the  time  t  or  if  (the  difference  between  which  may  be  re- 
garded as  infinitely  small); 

4th.  The  straight  line  b'a  gives  the  apparent  direction  for  the 
time  f,  freed  from  the  aberration  of  the  fixed  stars. 

Now,  the  points  P,  a,  bf  lie  in  a  straight  line,  and  the  portions 
Pa,  abf  will  be  proportional  to  the  intervals  of  time  t  —  T,  t'  —  t, 
if  the  velocity  of  light  is  uniform.  In  consequence  of  the 
immense  velocity  of  light,  the  interval  t'  —  T  will  always  be  so 
small  that  during  this  interval  we  may  suppose  the  motion  of 
the  earth  to  be  uniform  and  rectilinear;  consequently,  that  'A,  a, 
a'  also  lie  in  a  right  line,  and  the  portions  Aa,  aa'  are  also  pro- 
portional to  the  intervals  t  —  T,  t'  —  t.  Hence  it  follows  that 
the  lines  AP  and  b'a'  are  parallel,  and,  therefore,  that  the  first 
place  is  identical  with  the  third ;  that  is,  that  the  true  place  at  the 
time  T  is  identical  with  the  apparent  place  at  the  time  t. 

The  time  t  —  Twill  be  the  product  of  the  distance  Pa  into 
497*. 78,  which  is  the  time  in  which  light  describes  the  mean 
distance  of  the  earth  from  the  sun  (Art.  395),  this  mean  distance 
being  taken  as  the  unit.  In  this  computation  we  may  take  for 
the  distance  either  Pa  or  P^4.  or  pa,  without  sensible  difference 
in  the  resulting  value  of  t  —  T. 

From  these  principles  there  arise  three  methods  by  which  a 
planet's  (or  a  comet's)  apparent  place  may  be  found  from  the 
true  place  for  a  given  time  t : 

I.  From  the  given  time  t  we  subtract  the  time  required 
by  light  to  come  from  the  planet  to  the  earth.  We  thus 
obtain  the  reduced  time  T  for  which  the  true  place  is  iden- 
tical with  the  apparent  place  for  t. 

H.  The  true  place  and  the  distance  being  known  for  the 
time  /,  we  compute  the  reduction  /  —  T.  Thus,  by  means 
of  the  diurnal  motion  of  the  planet  (in  longitude  and  lati- 
tude, or  in  right  ascension  and  declination)  we  can  reduce 
the  true  place  from  the  time  t  to  the  time  T ';  and  the  true 
place  thus  found  is  the  apparent  place  at  the  time  t. 

III.  The  true  place  of  the  planet  at  the  time  T  as  seen 
from  the  point  in  which  the  earth  is  situated  at  the  time  t 
(or  the  direction  aP)  is  computed,  to  which  is  applied  the 


ANNUAL    PARALLAX.  643 

aberration  of  the  fixed  stars,  and  the  result  is  the  apparent 
place  at  the  time  £.* 

397.  If  a  and  3  are  the  true  right  ascension  and  declination 
of  a  planet  or  comet  at  a  time  £,  a/  and  S'  the  apparent  values 
for  the  same  time,  rf  its  distance  from  the  earth,  the  mean  dis- 
tance of  the.  earth  from  the  sun  being  unity,  A<X,  A£,  the  motion 
of  the  planet  in  right  ascension  and  declination  in  one  mean 
hour,  we  have,  according  to  the  method  II.  of  the  preceding 
article, 


in  which 

Kt  =  1?LZ§  log  k»  =  9.14073 

3600 

These  formulse  may  also  be  used  for  computing  the  sun's 
aberration  in  right  ascension  and  declination,  if  we  take  for  rf 
the  radius  vector  of  the  earth. 

HELIOCENTRIC   OR   ANNUAL   PARALLAX   OF   THE   FIXED    STARS. 

398.  The  heliocentric  parallax  of  a  star  is  the  difference 
between  its  true  places  seen  from  the  earth  and  from  the  sun. 
If  the  orbit  of  the  earth  were  a  circle  with  a  radius  equal  to  the 
mean  distance  from  the  sun,  the  maximum  difference  between 
the  heliocentric  and  geocentric  places  of  any  star  would  occur 
when  the  radius  vector  of  the  earth  was  at  right  angles  to  the 
line  drawn  from  the  earth  to  the  star.  This  maximum  corre- 
sponds, then,  to  the  horizontal  geocentric  parallax  ;  and  its  effect 
upon  the  apparent  places  of  stars  might  be  investigated  by  the 
methods  followed  in  Chapter  IV.  ;  but  we  prefer  to  employ  here 
the  method  just  exhibited  in  the  investigation  of  the  aberration, 
on  account  of  the  analogy  in  the  resulting  formulae. 

We  shall  call  the  maximum  angle  subtended  by  the  mean 
distance  of  the  earth  from  the  sun,  at  the  distance  of  the  star, 
the  constant  of  annual  parallax  of  the  star,  or,  simply,  its  annual 
parallax.  If  then  we  put 

*See  GAUSS,  Theoria  Motus  Corporum  Ccelestium,  Art.  71,  from  which  the  above 
article  is  chiefly  extracted.  Also,  for  the  application  of  method  III.,  see  the  same 
work,  Art.  118,  et  teq. 


644 


ANNUAL    PARALLAX. 


p  ~  the  annual  parallax, 

a  =  the  mean  distance  of  the  earth  from  the  sun, 

A  =  the  distance  of  the  star  from  the  earth, 


we  have 


or,  if  we  take  a  =  1,  according  to  the  usual  practice,  we  have, 
for  so  small  a  quantity, 

p  =  -    ^—rf  (688) 

To  find  the  'heliocentric  parallax  of  a  star  in  longitude  and 
latitude  at  a.  given  time,  the  annual  parallax  being  given. — 
Let  jT(Fig.  60)  be  the  place  of  the  earth  in  its  orbit, 
H  that  of  the  sun.  Conceive  a  plane  to  be  passed 
through  the  line  HT  and  a  star  S;  the  intersection 
of  this  plane  with  the  plane  of  the  ecliptic  is  the  line 
HT,  which,  produced  to  the  celestial  sphere,  meets 
it  in  a  point  E  of  the  ecliptic  whose  longitude  is  the 
earth's  heliocentric  longitude,  or  180°  +  O  (putting 
O  for  the  geocentric  longitude  of  the  sun  at  the 
H  given  time).  If  then  we  also  put 

r  —  the  distance  of  the  earth  from  the  sun  at  the  given  time, 
&  =  the  angle  SHE, 
$'=    «      «       STE, 

the  triangle  SHT  gives 


or 


sn  (#  _  0    = 


—  $  =  pr  sin 


(689) 


This  formula  corresponds  to  the  formula  (670)  for  the  aberra- 
tion reckoned  in  a  direction  from  a  point  (E)  of  the  ecliptic, 
only  in  the  present  case  this  point  is  in  longitude  180°  -f  O, 
while  in  the  case  of  the  aberration  it  was  in  longitude  90°  -j-  O. 
The  formulae  for  the  aberration  may  therefore  be  immediately 
applied  to  the  parallax  if  we  put  pr  for  k,  and  180°  -J-  O  for 
90°  +  O,  or  90°  +  O  for  O.  We  thus  find,  by  (675), 


pr  sin 
pr  cos 


—  O  )  sec 

—  O)  sin 


}    (690) 


REDUCTION    OF    STARS'    PLACES.  645 

400.  To  find  the  heliocentric  parallax  of  a  star  in  right  ascension 
and  declination,  the  annual  parallax  being  given. — By  (678),  putting 
pr  for  A',  and  90°  -j-  O  for  O,  we  have,  at  once, 

a'  —  a  =  —  pr  sec  3  (cos  O  sin  a  —  sin  Q  cos  e  cos  a)       ^ 
df  —  3  =  — pr  sin  Q  (cos  e  sin  d  sin  a  —  sin  e  cos  (5)  \    (691) 

—  pr  cos  O  si"  ^  cos  a  J 

401.  It  can  be  shown  from  (690)  that,  neglecting  the  small 
variations  produced  by  the  ellipticity  of  the  earth's  orbit,  the 
effect  of  the  annual  parallax,  considered  alone,  is  to  cause  the 
star  to  appear  to  describe  a  small  ellipse  about  its  mean  place 
in  one  sidereal  year ;  an  effect  entirely  analogous  to  that  of  the 
annual  aberration,  Art.  388.     But  the  maximum  and  minimum 
of  parallax  occur  when  the  earth  is  90°  from  the  points  at  which 
the  maximum  and  minimum  of  aberration  occur:  so  that  the 
major  axes  of  the  parallax  and  aberration  ellipses  are  at  right 
angles  to  each  other.     The  combined  effect  of  both  aberration 
and  parallax  is  still  to  cause  the  star  to  describe  an  ellipse,  the 
major  axis  of  which  is  equal  to  the  hypothenuse   of  a  right 
triangle,  of  which  the  two  legs  are  respectively  equal  to  the 
major  axes  of  the  two  ellipses.     For  this  combined  effect  is  ex- 
pressed by  the  following  formulae  (taking  r  =  1  for  a  circular 
orbit) : 

(A'  —  A)  =  —  [k  cos  (O  —  *)  —  P  sin  (O  —  >*)]  sec  ft 
(ft'  — ft)  =  —  [k  sin  (O  —  *)  +  p  cos(Q  —  *)]  sin  ft 

which,  if  we  assume  c  and  7*  to  be  determined  by  the  conditions 

c  sin  Y  =  k  sin  A  —  p  cos  A 
c  cos  y  =  k  cos  A  -f-  p  sin  A 
or 

c  sin  (A  —  Y)  —  P 
c  cos  (A  —  Y)  =  k 
become 

(A'  —  A)  =  —  c  cos  (O  —  Y)  sec  ft 
(ft'  — ft) c  sin  (O  —  r}  sin  ft 

in  which  we  have  c  =  \/(kz  +  p2).     This  form  for  the  total  effect 
is  entirely  analogous  to  that  for  the  aberration  alone. 

MEAN   AND    APPARENT    PLACES    OF   STARS. 

402.  The  formulae  above  given  enable  us  to  derive  the  appar- 
ent from  the  mean  place,  or  the  mean  from  the  apparent  place ; 


646  REDUCTION    OF    STARS*    PLACES. 

but  in  their  present  form  their  computation  is  exceedingly  trouble- 
some. We  owe  to  BESSEL  a  very  simple  arrangement  by  which 
their  application  is  facilitated. 

In  all  catalogues  of  stars  the  mean  places  only  can  be  given, 
and  these  only  for  a  certain  epoch.  For  each  star  there  is  given 
also  the  annual  precession  in  right  ascension  and  declination  :  so 
that  the  mean  place  for  any  time  after  or  before  the  epoch  of  the 
catalogue  is  readily  obtained,  as  in  the  example  of  Art.  374. 
But,  since  the  annual  precession  is  variable,  there  is  generally 
added  its  secular  variation,  which  is  the  variation  of  the  precession 
in  one  hundred  years.  Finally,  there  is  given  the  star's  proper 
motion. 

If  the  epoch  of  the  catalogue  is  <0,  and  the  mean  place  is  re- 
quired for  the  time  t.  and  we  denote  by 

p.  the  precession  for  the  epoch  t01 
Ap,  its  secular  variation, 
p.,  the  proper  motion, 

then,  since  in  computing  the  whole  precession  for  the  interval 
t  —  t0  we  must  employ  the  annual  precession  for  the  middle  of 
the  interval,  the  reduction  of  the  mean  place  to  the  time  t 
will  be 


This  form  applies  both  to  the  right  ascension  and  the  declination.* 
In  this  way  the  mean  place  is  brought  up  to  the  beginning  of 
any  given  year.  If  then  we  wish  the  apparent  place  for  a  time  T 
from  the  beginning  of  the  year,  T  being  expressed  in  fractional 
parts  of  the  year,  we  have  first  to  obtain  the  mean'place  for  the 
given  date  by  adding  the  precession  and  proper  motion  for  the 
interval  r,  and  then  the  apparent  place,  by  further  adding  the 
nutation  and  aberration.  Hence,  denoting  the  mean  right  ascen- 
sion and  declination  at  the  beginning  of  the  year  by  a  and  o,  the 
apparent  right  ascension  and  declination  for  the  given  time  r  by 


*  The  annual  proper  motions  being  also  variable  (Art.  379),  it  would  seem  that  their 
values  given  for  the  epoch  of  the  catalogue  could  not  be  carried  forward  to  another 
time  without  correction.  But,  to  avoid  the  necessity  for  this  separate  correction,  it 
may  be  included  in  the  secular  variation  of  the  precession,  as  is  done  by  ARGELAN- 
DER  in  his  catalogue,  "  DLX  Stellarum  Fixarum  Positiones  Medise,  ineunte  anno  1880." 


REDUCTION   OF   STARS     PLACES. 


647 


af  and  3f,  tne  annual  proper  motions  in  right  ascension  and  de- 
clination by  p.  and  /,  we  have,  by  (663),  (668),  and  (678), 


a'  =  a  -}-  T  (m  -f-  n  sin  a  tan  6)  -f-  rp 

—  (15".  81  48  +  6".8650  sin  a  tan  d)  sin  & 

15  .8321  6  .8682 

+  (  0  .1902  -f-  0  .0825  sin  a  tan  6)  sin  2ft 

—  (  0  .1872  +  0  .0813  sin  a  tan  6)  sin  2<£ 
-f-  (  0  .0621  -f  0  .0270  sin  a  tan  6)  sin  (C  — 

—  (  1  .1644  -f  0  .5055  sin  a  tan  d)  sin  20 
-f  (  0  .1173  -f  0  .0509  sin  a  tan  d)  sin  (Q  — 
-  (  0  .0195  -j-  0  .0085  sin  a  tan  <J)  sin  (Q  + 


—  9".2231  cos  a  tan  6  cos  ft 

9    .2240 
-f-  0  .0897  cos  a  tan  6  cos  2  & 

—  0  .0886  cos  a  tan  6  cos  2(£ 

—  0  .5510  cos  a  tan  d  cos  2  0 

—  0  .0093  cos  a  tan  6  cos  (Q  -f-  JT) 

20".4451  cos  £  cos  Q  cos  a  sec  6 
20  .4451  sin  Q  sin  a  sec  d 


(Precession  and  proper  motion.) 

for  1800 
1900 


1800 

1900 


(Nutation.) 


?•  (Aberration. ) 


1900 


d'  =  d  4-  r .  n  cos  a  -{-  r/  (Precession  and  proper  motion.) 

—  6".8650  cos  »  sin  ft    -f  9".2231  sin  a  cos  ft  for  1800 
6   .8682                                         9    .2240 

-f  0  .0825  cos  a  sin  2  ft  —  0  .0897  sin  a  cos  2  ft 

—  0  .0813  cos  a  sin  2  C  +0  .0886  sin  a  cos  2<£ 
-f-  0  .0270  cos  a  sin  (C  —  r') 

—  0  .5055  cos  a  sin  2  0  -f  0  .5510  sin  a  cos  2  0 
4-  0  .0509  cos  a  sin  (0  —  r) 

—  0  .0085  cos  a  sin  (Q  -f  I*)  -f-  0  .0093  sin  a  cos  (0  4- 


(Nutation.) 


20".  4451  cos  e  cos  Q  (tan  £  cos  6  —  sin  a  sin  6) 
20  .4451  sin  0  cos  a  sin  6 


|  (Aberration.) 


Now,  it  is  to  be  remarked  that  the  two  numerical  coefficients  of 
sin  £ ,  sin  2  ft ,  sin  23),  &c.  in  the  formula  for  a'  are  in  each 
case  very  nearly  in  the  ratio  of  m  to  n;*  and  hence,  if,  according 
to  the  method  of  BESSEL,  we  put 


6' 

'.8650  = 

ni 

15" 

.8148 

=  mi 

4-  h 

6 

.8682 

15 

.8321 

0 

.0825  = 

ni' 

0 

.1902 

=  mi' 

4-  h' 

0 

.0813  = 

ni" 

0 

.1872 

=  mi" 

4-  h" 

0 

.0270  = 

ni"' 

0 

.0621 

=  mi"' 

4-  h"' 

0 

.5055  = 

Hi" 

1 

.1644 

=  mi* 

4-  h* 

0 

.0509  = 

ni* 

0 

.1173 

=  mir 

4-  hr 

0 

.0085  = 

nt* 

0 

.0195 

=  mi* 

+  A" 

*  This  relation  is  not  accidental,  but  results  from  the  general  theory  of  nutation, 
which,  the  student  will  remember,  is  only  the  periodical  part  of  the  precession. 


648  REDUCTION   OF   STARS'    PLACES. 

we  shall  have 

*'  =  a  -f  [r  —  «  sin  £  -j-  i'  sin  2  £  —  i"sin  2£  -f  tv"sin  «[  —  r') 

—  eivsin  20  +  *vsin  (O  —  -r)  —  -*vi  sin  (O  +r)]  O  +  »  sin  a  tan 

—  [9".2231  cos  &  —  0".0897  cos  2  &  -f  0".0886  cos  2<C 

9  .2240 

+  0".5510  cos  2  O  +  0".0093  cos  (Q  -f  /*)]  cos  a  tan  d 

—  20".  4451  cos  e  cos  Q  cos  a  sec  6 

—  20  .4451  sin  Q  sin  a  sec  6 


—  A  sin  ££  -f-  A'  sin  2$  —  A"  sin  2£  +  A'"  sin  (<£—  r') 

_  Aiv  8in  2  O  +  ^v  sin  (Q  —  F  )  —  A"  sin  (Q  -j-  r} 

and 

6'  =  0  4-  [r  —  «  sin  £  -j-  t'  sin  2  &  —  i"  sin  2<£  +  i'"  sin  (([  —  7") 

—  i»v  gin  2  O  +  »v  sin  (Q  —  ^)  —  *'vi  sin  (  C  +  r  )]  n  cos  a 
-f  [9".2231  cos  £  —  0".0897  cos  2  &  +  0".0886  cos  2C 

9  .2240 

+  0".5510  cos  2Q  -f  0".0093  cos  (0  -f  r)]  sin  a 

—  20".4451  cos  e  cos  Q  (tan  e  cos  6  —  sin  a  sin  (J) 

—  20  .4451  sin  Q  cos  a  sin  6 


Putting  then,  in  accordance  with  BESSEL'S  original  notation, 
as  employed  in  the  American  Ephemeris  for  1865  and  subse- 
quent years, 

A  —       T  —  i  sin  &  +  l"  sin  2&  —  *"  sin  2  C  +  »"'  sin  (  C  —  -?*) 

—  ;*  sin  2  O  +  »v  sin  (0  —  r)  —  IM  sin  (  0  +  r) 
B  —  —  9".  2231  cos  &  +  0".0897  cos  2&  —  0".0886  cos  2(£ 

9   .2240 

—  0".5510cos  2  Q  —  0".0093  cos  (0  +  r) 
C  •=  —  20".4451  cos  t  cos  0 
D  =  —  20  .4451  sin  0 
E  =  —  h  sin  &  -f  h'  sin  2&  —  A"  sin  2£  +  A'"  sin  «[  —  /*) 

-  A'v  Sin  2  0  +  AV  sin  (0  —  r)  —  ATi  sin  (0  +  r) 

which  quantities  are  dependent  on  the  time,  and  are  wholly  inde- 
pendent of  the  star's  place  ;  and  also 

a  =  m  -f-  n  sin  a  tan  d  a'  =  n  cos  a 

b  =  cos  a  tan  d  bf  =  —  sin  a 

c  =  cos  a  sec  d  c'  =  tan  s  cos  d  —  sin  a  sin  d 

d  =  sin  a  sec  d  df  =  cos  a  sin  d 


REDUCTION    OF    STARS     PLACES. 


649 


which  depend  on  the  star's  place,  we  have 

o!  =  a  +  Aa  -f  Bb  -f  Cc  -f  Dd  -f  E  -f 
<5'  =  3  +  Aa'  +  Bb'  +  Cc'  -f  Dd!  +  r/*' 


}    (692) 


The  logarithms  of  A,  B,  (7,  D  are  given  in  the  Ephemeris  for 
every  day  of  the  year.  The  residual  E  never  exceeds  0".05,  and 
may  usually  be  omitted.  The  logarithms  of  a,  6,  c,  d,  a',  6',  c',  d' 
are  usually  given  in  the  catalogues,  hut  where  not  given  are 
readily  computed  by  the  above  formulse.  When  the  right  ascen- 
sion is  expressed  in  time,  the  values  of  a,  6,  c,  d,  above  given, 
are  to  be  divided  by  15. 

403.  If  we  substitute  the  values  of  m  and  n,  namely, 

for  1800,     m  =  46".0623  n  =  20".0607 

1900,     m  =  46  .0908  n  =  20  .0521 

we  find  the  following  values  of  z,  i7,  &c. : 


i 

i' 

i" 

i"' 

ff 

f 

;* 

1800 
1900 

0.34221 
0.34252 

0.00411 

0.00405 

0.00135 

0.02520 
0.02521 

0.00254 

0.00042 

h 

WT 

and  h',  h",  h'",  h*,  A*  inappreciable. 

1800 
1900 

-f  0".052 
+  0  .045 

-f  0".004 
+  0".003 

The  terms  in  f  and  ivi  in  the  expression  of  A  may  be  combined 
in  a  single  term ;  for,  putting 


^  have 


8n 


Jcos  J=      (P—  i*)  cos  r 
j  sin  J  =  —  O'r  +  f)  sin  /' 


—  T)  —  i"  sin  (O  +  H  =J  sin  (O  +  </) 


and  taking  for  1800,  P=  279°  30'  8r/;  and  for  1900,  F=  281° 
1?'  42",  we  find 


1800 
1900 

y 

J 

+  0.00294 
-f-  0.00293 

83°  10' 
81  55 

650  REDUCTION    OF    STARS'    PLACES. 

Hence  the  values  of  J.,  .B,  and  E  may  be  expressed  as  follows  : 

A  _  T  _  0.34221  sin  £      —  0.02520  sin  2Q  +  0.00294  sin  (Q  -f  83°  10')  for  1800 
0.34252  0.00293  81      55         "1900 

-f  0.00411  sin  2^  —  0.00405  sin  2£  +  0.00135  sin  (£  —  r') 

B  ,  =     _  9".2231  cos  &      —  0".5510  cos  2  Q  —  0".0093  cos  (Q  +  279°  30')"    1800 
9  .2240  281     13     "      1900 

+  0  .0897  cos  2  £  —  0  .0886  cos  2<[ 
E=     —  0".052    shift     —  0".004    sin2O  "    180° 

0  .045  0  .003  "      1900 

These  values  agree  (within  quantities  practically  inappreciable) 
with  those  given  by  Dr.  PETERS  (Numerus  Constans  Nutationis,  pp. 
75,  76).  It  is  necessary  to  remark  that  in  the  British  Associa- 
tion Catalogue  and  the  British  Nautical  Almanac,  the  preceding 
values  of  C  and  D  are  denoted  by  A  and  B,  and  vice  versa.* 
See  p.  94. 

404.  When  the  catalogue  does  not  give  the  logs  of  a,  b,  c,  &c., 
another  form  of  reduction,  also  proposed  by  BESSEL,  may  some- 
times be  preferred.  By  putting 

/  =  mA  +  E  i=  C  tans 

g  cos  G-  =  nA  h  cos  If  =  D 

g  sin  &'=  B  h  sin  H  =  C 

we  find 


a'=o-|-/ 

d'=d~\-ico&d-{-TfS-\-  gcos(G  +  a)  -f  h  cos  (H  -f  a)  sin  3    j  * 


The  values  of  /,  log  g,  G,  log  A,  H,  log  i,  and  log  r  are  given  in 
the  Ephemeris  for  every  day  of  the  year. 

* 

405.  A  star's  apparent  place  may  be  reduced  to  its  mean  place 
and  referred  to  the  mean  equinox  of  any  given  date  by  reversing 
the  signs  of  the  reductions  as  above  determined.  By  the 
apparent  place  of  a  star  we  here  mean  the  apparent  geocentric  place. 
The  observed  place  (that  seen  from  the  surface  of  the  earth)  differs 

*  This  interchange  of  letters,  most  unnecessarily  introduced  by  BAILY  in  the  British 
Association  Catalogue,  produces  considerable  inconvenience,  as  in  most  of  the  Euro- 
pean catalogues  of  stars  BESSEL'S  notation  is  preserved,  while  in  the  English  Almanac 
BAILY'S  notation  is  followed.  In  the  American  Ephemeris  for  1865  and  subsequent 
years  the  notation  of  BESSEL  has  been  restored  :  an  example  which  will  doubtless  be 
followed  by  the  British  Almanac  at  an  early  day. 


FICTITIOUS    YEAR.  651 

from  this  by  the  diurnal  aberration  and  the  refraction ;  but  the 
first  of  these  corrections  depends  on  the  latitude  of  the  observer 
and  the  star's  hour  angle,  and  the  second  upon  the  star's  zenith 
distance  :  so  that  neither  of  them  can  be  brought  into  the  com- 
putation of  a  star's  position  until  the  place  of  observation  and 
the  local  time  are  given. 

406.  The  fictitious  year. — In  the  preceding  investigations,  we 
have  used  the  expression  "  beginning  of  the  year,"  without  giving 
it  a  definite  signification.  For  the  purpose  of  introducing 
uniformity  and  accuracy  in  the  reduction  of  stars'  places,  BESSEL 
proposed  a  fictitious  year,  to  begin  at  the  instant  when  the  sun's 
mean  longitude  is  280°.  This  instant  does  not  correspond  to 
the  beginning  of  the  tropical  year  on  the  meridian  of  Greenwich*; 
that  is,  the  (mean)  sun  is  not  at  this  instant  on  the  meridian  of 
Greenwich,  but  011  a  meridian  whose  distance  from  that  of 
Greenwich  can  always  be  determined  by  allowing  for  the  sun's 
mean  motion.  This  meridian  at  which  the  fictitious  year  begins 
will  vary  in  different  years;  but,  since  the  sun's  mean  right 
ascension  is  equal  to  his  mean  longitude  (Art.  41),  the  sidereal 
time  at  this  meridian  when  the  fictitious  year  begins  is  always 
18*  40M  (=  280°).  By  the  employment  of  this  epoch,  therefore, 
the  reckoning  of  sidereal  time  from  the  beginning  of  the  year  is 
simplified,  and,  accordingly,  it  is  now  generally  adopted  as  the 
epoch  of  the  catalogues  of  stars.  In  the  value  of  log  A,  which 
involves  the  fraction  of  a  year  (r),  the  same  origin  of  time  must 
be  used ;  and  this  is  attended  to  in  the  computation  of  the  Ephe- 
merides,  which  now  give  not  only  the  logarithms  of  A,  _B,  (7, 
and  D,  but  also  the  value  of  r  (or  its  logarithm)  reckoned  from 
the  beginning  of  the  fictitious  year  and  reduced  to  decimal  parts 
of  the  mean  tropical  year. 

For  all  the  purposes  of  reduction  of  modern  observations,  the 
computer  need  not  enter  further  into  this  subject,  and  may 
depend  upon  the  Ephemerides.*  But,  as  the  subject  is  inti- 

*  The  reduction  of  observations  made  between  1750  and  1850  will  be  most  con- 
veniently performed  by  the  aid  of  the  Tabulse  Regiomontanse  of  BESSEL.  The  con- 
stants used  by  BESSEL  differ  materially  from  those  now  adopted  in  the  American  and 
British  Almanacs.  Professor  HUBBARD  has  given  a  very  simple  table  by  which  the 
values  of  log  A,  log  B,  log  (7,  and  log  D  as  given  in  the  Tab.  Reg.  may  be  reduced 
to  those  which  follow  from  the  use  of  PETERS'S  constants,  in  the  Astronomical  Journal, 
Vol.  IV.  p.  142.  The  special  and  general  tables  for  the  reduction  of  stars'  places, 


652  LENGTH    OF    THE   YEAR. 

matety  connected  with  that  of  time  in  general,  I  shall  prosecute 
it  a  little  further. 

407.  The  sun's  mean  motion,  and  the  length  of  the  year. — Accord- 
ing to  BESSEL,*  the  sun's  mean  longitude  at  mean  noon  at  Paris 
in  1800,  January  0,  is 

279°  54'  1".36 

and  the  sun's  sidereal  motion  in  365.25  mean  days  is 

360°  —  22".617656 

(By  January  0  is  denoted  the  noon  of  December  31  in  the  com- 
mon mode  of  expressing  the  date ;  and,  consequently,  Jan.  1,  2, 
&c.  denote  1  day,  2  days,  &c.  from  the  epoch,  while  in  the  com- 
mon mode  they  mean  the  beginning  of  the  1st  day,  2d  day,  &c.) 
The  sidereal  motion  is  referred  to  &  fixed  point  of  the  ecliptic; 
but  the  mean  longitude  is  referred  to  the  moving  vernal  equinox. 
Hence  the  change  of  the  mean  longitude  in  any  time  is  the 
sidereal  motion  in  that  time  plus  the  general  precession ;  and 
therefore,  adopting  here  BESSJEL'S  precession  constant,  f  in  order 
to  follow  his  computations, 

Sid.  motion  in  365d.25     =  360°  —  22".617656 

General  precession          =  -f  60  .22350    -f  0".000244361g 

Mean  motion  in  365d.25  =  360°  -f  27  .605844  -f-  0  .000244361* 

and,  dividing  by  365.25, 

Mean  daily  motion  =  59'  8".3302-f-  0".0000006902f 

where  t  is  the  number  of  years  after  1800.  The  secular  change 
of  the  mean  motion,  expressed  by  the  second  term,  brings  with 
it  a  secular  change  of  the  length  of  the  tropical  year.  This  year 

given  in  the  Washington  Astronomical  Observations,  Vol.  III.,  Appendix  C,  are  also 
adapted  to  the  new  constants. 

For  the  reduction  of  observations  from  1850  to  1880,  the  Tab.  Reg.  have  been 
continued  by  WOLFERS  and  ZECH  (Tabulx  Reductionum  Observationum  Astronomicarum 
Annis  1860  usque  ad  1880  respondentes,  auctore  J.  PH.  WOLFERS  :  Additse  sunt,  Tabulse 
Regiomontanse,  annis  1850  usque  ad  1860  respondentes  ab  ILL.  ZECH  continuatee.  Berlin, 
1850).  In  the  continuation  by  ZECH,  which  extends  from  1850  to  1860,  all  the 
constants  are  the  same  as  those  used  by  BESSEL  ;  in  the  continuation  by  WOLFERS, 
from  1860  to  1880,  BESSEL'S  precession  constant  is  retained,  but  PETERS'S  nutation 
constant  is  adopted. 

*A*tron.  Nock.,  No.  133.  f  Ibid. 


LENGTH    OF   THE    YEAR.  653 

is  the  time  in  which  the  sun  changes  his  mean  longitude  exactly 
360°,  and  is,  therefore,  found  by  dividing  360  by  the  mean  daily 
motion  :  thus,  if  we  put 

Y  =  the  length  of  the  tropical  year  in  mean  solar  days, 
we  find 

Y=  365^.242220027  —  <K00000006886f 

where  the  value  of  the  second  term  for  t  =  100  is  08.595,  which 
is  the  diminution  of  the  length  of  the  tropical  year  in  a  century. 
The  length  of  the  sidereal  year  is  invariable,  and  is  readily 
found  by  adding  to  365.25  the  time  required  by  the  sun  to  move 
through  22". 617656  at  the  rate  of  his  sidereal  motion;  or,  putting 

Y'  =  the  length  of  the  sidereal  year, 
by  the  proportion 

360°  —  22".617656  :  360°  =  365*.25  :  T 

which  gives 

Y'  =  365.256374416  mean  solar  days, 
==  366.256374416  sidereal  days. 

408.  The  epoch  of  the  sun's  mean  longitude. — This  term  denotes 
the  instant  at  which  the  common  year  begins.  The  value  of  the 
longitude  itself  at  this  instant  is  frequently  called  "the  epoch," 
and  is  denoted  by  E.  Its  value  for  January  0  of  any  year, 
1800  -{-  t,  is  found  by  adding  the  motion  in  365  days  for  each 
year  not  a  leap  year,  and  the  motion  in  366  days  for  each  leap 
year.  The  motion  in  365  days  is  foun4  from  the  above  value 
for  365.25  days  by  deducting  one-fourth  the  mean  daily  motion, 
or  14'  47".083 :  so  that,  if  /  denotes  the  remainder  after  the 
division  of  t  by  4,  we  have,  for  the  epoch  of  1800  +  t,  Jan.  0,  at 
Paris, 

#=279°  54'  1".36  +  27".605844£  +  0".0001221805*2 

-  (14'  47".083)/  (693) 

To  extend  this  formula  to  years  preceding  1800,  we  must  put 
/—  4  in  the  place  of  /:  so  that  the  multiplier  of  (—  14'  47".083) 
will  be,  for  example,  —  1,  —  2,  —  3,  —  4,  —  1,  &c.  for  the  years 
1799,  '98,  '97,  '96,  '95,  &c.  But  these  rules  for  /  will  give  the 


654  THE    YEAR. 

mean  longitude  at  the  beginning  of  the  leap  years  too  great  by 
the  motion  in  one  day  (since  the  additional  day  is  not  added  until 
the  end  of  February) ;  and  therefore  the  epoch  for  these  years 
is  January  1  instead  of  January  0.  A  general  table  containing 
the  mean  longitude  at  mean  noon  for  every  day  of  the  year, 
computed  from  the  mean  longitude  for  Jan.  0  by  the  formula, 
will  be  applicable  to  leap  years  if  in  the  months  of  January  and 
February  we  increase  the  argument  of  the  table  by  one  day,  as 
in  Table- VI.  of  the  Tab.  Reg. 

409.  To  find  the  beginning  of  the  fictitious  year. — Denoting  the 
mean  time  from  the  beginning  of  the  fictitious  year  to  Jan.  0  of 
any  year  by  A:,  we  have 

E  —  280° 
mean  daily  motion 

whence,  taking  the  daily  motion  =  59'  8". 3302,  we  find,  in  deci- 
mals parts  of  a  mean  day, 

k=  —  0.10107289  +  0.0077799535 1 

-\f     +  0.000000034433  P 

The  quantity  k  is  evidently  equal  to  the  east  longitude  from 
Paris  of  that  terrestrial  meridian  on  which  the  fictitious  year 
begins  (Art.  406). 

410.  In  the  Tabula?  Regiomontance  the  argument  is  the  reduced 
date  as  it  would  be  reckoned  at  the  meridian  in  the  east  longitude 
A,  the  beginning  of  the  fictitious  year  being  always  denoted  by 
January  0.     If  then  d  is  the  west  longitude  from  Paris  of  any 
place,  the  instant  of  mean  noon  at  this  place  corresponds  to  the 
instant  k  -f-  d  of  the  fictitious  meridian,  and  therefore  k  -f-  d  is 
the  reduction  to  apply  to  the  mean  time  at  the  place  to  obtain 
the  argument  with  which  to  enter  those  tables. 

But,  if  the  sidereal  time  at  the  place  d  is  given,  it  is  most  ex- 
pedient to  reckon  the  time  at  once  in  sidereal  days  from  the 
beginning  of  the  fictitious  year.  Accordingly,  in  the  tables  con- 
taining the  values  of  log  A,  log  J5,  &c.  for  the  reduction  of  stars, 
the  argument  is  the  sidereal  date  at  the  fictitious  meridian.  To 
obtain  this  date,  it  is  to  be  observed,  first,  that  the  tables  are  im- 
mediately available  on  the  fictitious  meridian  for  the  sidereal 
time  18A  40™,  without  any  reduction  of  the  date.  For  any  other 


SIDEREAL   TIME.  655 

meridian,  at  the  sidereal  time  18*  40m  the  argument  of  the  table 
will  be  the  reduced  date  ;  but  at  any  other  sidereal  time  g  the 
argument  must  be  this  reduced  date  increased  "by 


24* 
which  must  be  always  taken  <  1  and  positive  ;  or  by  the  quantity 

g  _|_  5*  20*" 


9'  = 


24* 


omitting  one  whole  day  if  g  -f  5*  20™  >  24*.  Now,  in  order  that 
the  local  date  may  correspond  with  that  supposed  in  the  tables, 
the  day  must  be  supposed  to  begin  at  the  instant  when  that  point 
is  on  the  meridian  whose  right  ascension  is  18*  40™.  Therefore, 
whenever  the  right  ascension  of  the  sun  is  as  great  as  18*  40W, 
so  that  the  point  in  question  culminates  before  the  sun,  one  day 
must  be  added  to  the  common  reckoning.  Hence  the  formula 
for  preparing  the  argument  of  the  tables  will  be 

Argument  =  Eeduced  date  -f-  gr  -f  i; 

in  which  we  must  take  i  =  0  from  the  beginning  of  the  year 
to  the  time  when  the  sun's  R.  A.  =  g,  and  i  =  -f-  1  after  this 
time. 

The  values  of  gf  are  given  on  p.  16  of  the  Tab.  Reg.  for  given 
values  of  g.  The  values  of  k  are  given  in  Table  I. 

The  values  of  log  A,  log  B,  log  (7,  log  D  are  also  given  in  the 
Berlin  Jahrbuch  for  the  fictitious  date  ;  and  the  constants  of  pre- 
cession, nutation,  and  aberration  are  the  same  as  those  employed 
by  BESSEL  in  the  Tab.  Reg. 

411.  Conversion  of  mean  into  sidereal  time,  and  vice  versa. — In  the 
explanation  of  this  subject  in  Chapter  II.  we  said  nothing  of  the 
effect  of  nutation,  which  we  will  now  consider.  Let  us  go  back 
to  the  definitions  and  state  them  more  precisely. 

1st.  The  first  mean  sun,  which  may  be  denoted  by  Ot,  moves 
uniformly  in  the  ecliptic,  returning  to  the  perigee  with  the  true 
sun.  The  longitude  of  this  fictitious  sun  referred  to  the  mean 
equinox  is  called  the  sun's  mean  longitude. 

2d.  The  second  mean  sun,  which  may  be  denoted  by  O2,  moves 


656  SIDEREAL    TIME. 

uniformly  in  the  equator,  returning  to  the  mean  equinox  with  the 
first  mean  sun. 

3d.  The  sidereal  day  begins  with  the  transit  of  the  true  equi- 
nox ;  and  the  sidereal  time  is  the  hour  angle  of  the  true  equinox. 

Hence  it  follows  that 

the  mean  R.  A.  of  O3  =  the  mean  long,  of  QI  =  the  sun's 
mean  longitude; 

and  since  when  O2  is  on  the  meridian,  its  R.  A.  reckoned  from 
the  true  equinox  is  also  the  hour  angle  of  the.  true  equinox,  it 
also  follows  that 

*  "F0  =  the  sidereal  time  at  mean  noon. 
=  true  E.  A.  of  03 

=  mean  E.  A.  of  Qa  +  nutation  of  the  equinox  in  E.  A. 
=  sun's  mean  longitude  -j-  nutation  of  the  equinox  in  R.  A. 

The  nutation  of  the  equinox  in  R.  A.  is  found  from  the  first 
equation  on  p.  626  by  putting  a  =  0,  S  =  0,  whence 

nutation  of  equinox  in  R.  A.  =  AA  cos  e 

which  is  the  quantity  given  in  the  Nautical  Almanac  as  the 
"equation  of  the  equinoxes  in  right  ascension." 

Since  the  nutation  is  contained  in  the  value  of  V0  given  in  the 
Almanac  for  each  mean  noon,  no  further  attention  to  it  is  needed 
when  that  work  is  consulted ;  and  the  rules  given  in  Chapter  II. 
are  therefore  practically  complete. 

For  the  conversion  of  time  between  1750  and  1850,  the  Tab., 
Reg.  furnish  the  following  facilities : — Table  VI.  gives  the  right 
ascension  of  the  second  mean  sun,  corrected  for  the  solar  nuta- 
tion of  the  equinox,  for  every  mean  noon  at  the  fictitious  meri- 
dian /.:.  Since  the  fictitious  year  always  begins  with  the  same 
mean  longitude  of  the  sun  (or  right  ascension  of  O2),  the  num- 
bers of  this  table  are  general,  and  may  be  used  for  every  year. 
The  number  taken  from  this  table  for  any  given  date  (which 
must  be  the  reduced  date  above  explained)  are  then  corrected  for 
the  lunar  nutation  of  the  equinox  in  right  ascension,  which  is 
given  in  Table  IV.  for  all  dates  between  1750  and  1850.  We 
thus  obtain  the  sidereal  time  at  mean  noon  (=  V0)  at  the  fictitious 
meridian  on  the  given  day.  Any  given  mean  time  at  another 
meridian  is  then  converted  into  the  corresponding  sidereal  time, 


REDUCTION  OF  A  PLANET'S  PLACE.  657 

or  vice  versa,  according  to  the  rules  in  Chapter  II.,  employing  the 
V0  for  the  fictitious  meridian  precisely  as  it  was  there  employed 
for  the  meridian  of  Greenwich. — The  longitude  of  the  place  to 
he  used  here  is  k  -f-  d,  d  heing  the  west  longitude  of  the  place 
from  Paris,  and  k  the  east  longitude  of  the  fictitious  meridian 
from  Paris  given  in  Tahle  I. 

REDUCTION  OF  THE  APPARENT  PLACE  OF  A  PLANET  OR  COMET. 

412.  The  observed  place  of  a  planet  (or  comet)  heing  freed 
from  the  effect  of  refraction,  diurnal  aberration,  and  geocentric 
parallax,  we  have  the  apparent  geocentric  place,  referred  to  the 
true  equator  and  equinox  of  the  time  of  observation,  and  affected 
by  the  planetary  aberration.  For  the  calculation  of  a  planet's 
orbit  from  three  or  more  observations  at  different  times,  it  is 
necessary  to  refer  its  places  at  these  times  to  the  same  common 
fixed  planes,  which  is  most  readily  effected  by  reducing  all  the 
places  to  the  equinox  of  the  beginning  of  the  year  in  which  the 
observations  are  made,  or,  when  the  observations  extend  beyond 
one  year,  to  the  beginning  of  any  assumed  year.  To  effect  this, 
we  must  apply  to  each  apparent  geocentric  place — 1st.  The  aber- 
ration (687),  with  its  sign  reversed,  in  computing  which  the  posi- 
tion of  the  observer  on  the  surface  of  the  earth  may  be  con- 
sidered by  taking  r'  equal  to  the  actual  distance  of  the  planet 
from  the  observer  at  the  time  of  observation.  This  distance  is 
found  from  the  geocentric  distance  at  the  same  time  with  the 
parallax,  by  the  equation  (137). 

2d.  The  nutation  for  the  date  of  the  observation,  with  its 
sign  reversed. 

3d.  The  precession  from  the  date  of  the  observation  to  the 
assumed  epoch,  which  will  be  subtracted  or  added  according  as 
the  epoch  precedes  or  follows  the  date. 

But  the  nutation  and  precession  are  most  conveniently  com- 
puted together  by  the  aid  of  the  constants  A  and  B  used  for  the 
fixed  stars.  These  constants  being  taken  for  the  date,  a,  6,  a', 
and  b'  are  to  be  computed  as  in  Art.  402,  with  the  right  ascen- 
sion and  declination  of  the  planet ;  and  then  to  the  a  and  <?, 
already  corrected  for  aberration,  we  apply  the  corrections  —  (Aa 
-f  Bb)  and  —  (Aa!  -f  Bb'}  respectively.  The  place  thus  obtained 
is  the  true  place  of  the  planet  referred  to  the  mean  equinox  of  the 
beginning  of  the  year.  If  the  several  observations  are  in  different 

VOL.  I.— 42 


658  OBLIQUITY   OF   THE   ECLIPTIC. 

years,  they  are  then  to  be  reduced  to  the  same  epoch  by  simply 
applying  the  annual  precession,  c  being  the  annual  precession  in 
right  ascension,  and  c'  that  in  declination. 

When  the  distance  of  the  planet  is  not  known,  the  aberration 
is  taken  into  account  by  Method  III.  of  Art.  896 ;  but  the  details 
of  this  subject  belong  to  the  computation  of  orbits,  which  is  re- 
served for  Physical  Astronomy. — See  GAUSS,  Theor.  Mot.  Corp. 
Ccel.y  Art.  118  et  seq. 


CHAPTER   XII. 

DETERMINATION  OF  THE  OBLIQUITY  OF  THE  ECLIPTIC  AND  THE 
ABSOLUTE  RIGHT  ASCENSIONS  AND  DECLINATIONS  OF  STARS  BY 
OBSERVATION. 

413.  THE  most  obvious  method  of  finding  the  obliquity  of  the 
ecliptic  is  to  measure  the  sun's  apparent  declination  at  either  the 
northern  or  the  southern  solstice  ;  for  at  these  points,  assuming 
the  sun  to  be  exactly  in  the  ecliptic,  the  declination  is  equal  to 
the  obliquity.     Indeed,  without  any  reference  to  the  sun's  abso- 
lute declination,  a  rude  approximate  value  of  the  obliquity  is  at 
once  derived  by  taking  one-half  of  the  difference  of  the  meridian 
altitudes  of  the  sun  on  the  21st  of  June  and  the  21st  of  December. 
Upon  this  principle  the  ancients  succeeded  in  measuring  the 
obliquity  by  observing  the  greatest   and   least   lengths  of  the 
meridian  shadow  of  a  gnomon. 

414.  In  what  follows,  we  suppose  the  sun's  declination  to  be 
observed.     This  is  obtained  from  the  true  meridian  zenith  dis- 
tance (£)  of  the  sun's  centre,  and  the  known  latitude  of  the  place 
of  observation  (^),  by  the  formula* 


*  The  sign  of  f  is  to  be  changed  when  the  sun  is  north  of  the  zenith  of  the 

observer. 


OBLIQUITY    OF    THE    ECLIPTIC.  659 

415.  Now,  the  sun's  declination  is  equal  to  the  obliquity  only 
when  it  has  reached  its  maximum  (northern  or  southern)  limit, 
that  is,  precisely  at  the  solstitial  points.     But,  since  the  sun  will, 
in  general,  not  arrive  at  the  solstice  at  the  same  time  that  it 
culminates  at  the  particular  meridian  at  which  the  observation  is 
made,  we  cannot  directly  measure  this  maximum  by  meridian 
observations.     But  we  can  measure  the  declination  at  several 
successive  transits  near  the  solstice,  and  then  by  interpolation 
infer  the  maximum  value.     A  simpler  practical  process  (which 
we  shall  explain  fully  below)  is  to  reduce  each  observation  to  the 
solstice ;  but  this  requires  us  to  know  (at  least  approximately) 
the  time  when  the  sun  arrives  at  the  solstice,  and  this,  again, 
supposes  a  knowledge  of  the  position  of  the  equinoctial  points, 
which  are  90°  distant  from  the  solstitial  points. 

The  position  of  the  equinoctial  points  may  be  determined  by 
observing  the  sun's  declination  on  several  successive  days  near 
the  time  of  the  equinoxes,  and,  by  interpolation,  finding  the  time 
when  the  declination  is  zero.  At  the  same  time,  a  comparison 
must  be  made  between  the  times  of  transit  of  the  sun  and  some 
star,  adopted  as  &  fundamental  star:  so  that  the  distance  of  the 
star  from  the  equinoctial  point,  or  its  right  ascension,  is  fixed. 
We  may  then  regard  the  star  as  a  fixed  point  of  comparison  by 
which  the  instants  when  the  sun  arrives  at  any  given  points  (as 
the  solstices)  may  be  determined.  But,  instead  of  finding  the 
equinoctial  points  by  a  direct  interpolation,  it  is  preferable  in 
this  case  also  to  refer  each  observation  to  the  equinox,  which,  as 
will  be  seen  below,  requires  an  approximate  knowledge  of  the 
obliquity  of  the  ecliptic. 

The  determination  of  these  two  elements,  the  obliquity  of  the 
ecliptic  and  the  position  of  the  equinoctial  points,  is,  therefore, 
effected  by  successive  approximations ;  but,  in  the  actual  state 
of  astronomy,  the  approximations  are  already  so  far  carried  out 
that  the  remaining  error  in  either  element  can  be  treated  as  a 
differential  which,  by  a  judicious  arrangement  of  the  observations, 
produces  only  insensible  errors  of  a  higher  order  in  the  other 
element.  I  proceed  to  treat  fully  of  the  precise  practical 
methods. 

416.  Determination  of  the  obliquity  of  the  ecliptic. — Let  D  be  the 
sun's  apparent  declination  derived  from  an  observation  near  the 
solstice ;  A  its  apparent  right  ascension  at  the  time  of  the  obser- 


660  OBLIQUITY    OF   THE   ECLIPTIC. 

vation,  derived  from  the  solar  tables ;  e  the  apparent  obliquity 
of  the  ecliptic  for  the  same  time.  If  the  sun  were  exactly  in 
the  ecliptic,  we  should  have,  by  (34), 

Bin  A  tan  e  =  tan  D 

but  accuracy  requires  that  the  sun's  latitude,  /5,  should  be  taken 
into  account.  We  have,  by  (29), 

sin  /? 
tan  D  —  tan  e  sm  A  — 


cos  D  cos  e 
which,  if  we  put 

tan  D'  —  tan  e  sin  A  (694) 

becomes 

sin(D  — D')  sin/3 

tan  D —  tan  D'  = 


cos  D  cos  D'       cos  D  cos  s 
whence,  with  sufficient  accuracy,  since  /9  never  exceeds  1", 

D  —  D'  =  ft  sec  s  cos  D  (695) 

Hence,  if  the  correction  /?  sees  cos  D  is  subtracted  from  the 
given  declination  Z),  we  shall  obtain  the  reduced  declination  D> ', 
from  which,  by  (694),  we  can  deduce  e.  It  is  evident  that  D'  is 
the  declination  of  the  point  in  which  the  ecliptic  is  intersected 
by  the  declination  circle  drawn  through  the  sun's  centre,  and 
we  may  call  the  quantity  /?  sec  s  cos  D  the  reduction  to  the  ecliptic. 
Near  the  solstices,  however,  this  reduction  does  not  sensibly 
differ  from  /?,  since  cos  e  and  cos  D  are  then  very  nearly  equal. 
We  shall,  therefore,  in  the  present  problem,  find  the  reduced 
declination  by  the  formula  D'  =  D  —  /9 ;  and  then  we  have,  by 
(694), 

tan  e  =  tan  D'  cosec  A  (696) 

Instead  of  computing  e  from  this  equation  directly,  it  is  usual 
to  employ  its  development  in  series  by  which  the  difference  of 
e  and  Df  is  obtained.  For,  since  A  near  the  northern  solstice  is 
nearly  90°,  if  we  put 

u  =  90°  —  A 

u  will  be  a  small  angle  whose  cosine  and  secant  will  not  differ 
much  from  unity,  and  the  equation  (696),  expressed  in  the  form 


OBLIQUITY    OF    THE    ECLIPTIC.  661 

tan  Df  =  tan  e  cos  w,  will  be  developed  in  the  series  [PL  Trig., 
Art.  254] 

D'  —  e  =  q  sin  2  e  -f  \(f  sin  4  e  -f  \  q3  sin  6  e  -f  &c. 

in  which 

cos  u  —  1 

q  =  -  -  =  —  tan2  £  u 

cos  u  -f-  1 

and  the  terms  of  the  series  are  expressed  in  arc.     Reducing  to 
seconds,  and  putting 

x  =  the  reduction  to  the  solstice, 
or 

x  =  ^  sin  2  .  -' t-^  sin  4  e  +  fto.  (697) 

sin  1"  "sin  1 " 

we  have,  at  the  northern  solstice, 

s  =  Df  -\-  x=D  —  ft  +  x  (698) 

The  reduction  x  can  be  tabulated,  for  any  assumed  value  of  e, 
with  the  argument  u.  The  changes  of  the  tabular  numbers 
depending  on  a  change  of  the  obliquity  may  also  be  given  in  the 
table :  so  that  these  numbers  may  be  readily  made  to  correspond 
to  any  assumed  obliquity. 

For  the  southern  solstice,  we  take  u  =  270°  —  A,  and  the 
equation  (696)  will  give  tan  D*  =  —  tan  e  cos  M,  the  development 
of  which  gives  the  algebraic  sum  Df  -\-  e ;  but  we  can  avoid  the 
use  of  two  formulae  by  throwing  this  change  of  sign  upon  e, 
regarding  the  obliquity  obtained  from  the  southern  solstice  as 
negative,  during  the  computation.  This  simply  changes  the  sign 
of  the  reduction  x. 

417.  Let  us  now  inquire  what  effect  an  error  in  the  right 
ascensions  taken  from  the  tables,  or  in  M,  will  produce  in  the 
computed  value  of  e.  Differentiating  the  equation  (696)  with 
reference  to  e  and  A  =  ±  90°  —  w,  we  find 

ds  =  ?  tan  u  sin  2s  du 

If  we  suppose  the  error  in  the  tabular  right  ascension  of  the 
sun  to  be  in  any  case  as  great  as  one  second  of  time  (the  actual 
probable  error,  however,  being  much  less),  and,  therefore,  sub- 
stitute in  this  equation  du  =  15",  e  =  23°  27'.5,  we  find 

ds  =  5".48  tan  u 


662  OBLIQUITY    OF    THE    ECLIPTIC. 

For  u  —  10°,  this  gives  de  =  0".97.  The  sun's  motion  being 
about  1°  per  day,  we  shall  have  u  <  10°  for  observations  within 
ten  days  of  the  solstice,  and  the  error  in  the  computed  obliquity 
less  than  1",  even  if  the  error  in  the  right  ascensions  is  as  great 
as  15".  But  this  error  will  be  wholly  eliminated  if  observations 
equidistant  from  the  solstice  preceding  and  following  it  are  com- 
bined; for  then  u,  and  consequently  also  ds,  will  have  equal 
numerical  values  with  opposite  signs,  and  the  errors  will  destroy 
each  other  in  the  mean. 

418.  The  mean  of  the  values  of  the  obliquity  found  from  a 
number  of  observations,  preceding  and  following  the  solstice 
and  symmetrically  disposed,  will,  therefore,  be   taken   as   the 
value  of  the  obliquity  at  the  time  of  the  solstice,  free  from  errors 
in  the  right  ascension,  and  affected   only  by  the  unavoidable 
errors  of  observation  and  by  any  errors  that  may  exist  in  the 
refraction  and  parallax  or  in  the  latitude  of  the  place  of  observa- 
tion.   The  error  in  the  latitude  is  eliminated  by  taking  the  mean 
of  the  values  of  the  obliquity  found  at  the  northern  and  the 
southern  solstices.    The  error  of  the  refraction  tables  will  at  the 
same  time  be  partially  eliminated ;  but  not  wholly,  since  these 
errors  have  probably  different  values  at  zenith  distances  differing 
so  much  as  47°  ;  but  a  sensible  error  in  the  mean  resulting  from 
any  probable  error  in  the  present  value  of  the  solar  parallax  is 
not  to  be  feared. 

Before  taking  the  mean,  however,  it  is  proper  to  deduct  from 
each  value  the  nutation  of  the  obliquity  (AS,  Art.  381),  for  the 
times  of  the  two  solstices  respectively,  whereby  we  obtain  the 
mean  obliquity ;  and  then  to  reduce  this  to  the  same  fixed  epoch, 
as  the  beginning  of  the  year,  by  allowing  for  the  animal  decrease. 
The  value  of  this  annual  decrease  adopted  in  (646)  is  0".4738 ; 
but  this  value  was  deduced  by  PETERS  from .  theory,  while  the 
value  derived  directly  from  observations  at  distant  periods  is, 
according  to  BESSEL,  0".457,  and,  according  to  PETERS,  0".4645. 

In  combining  a  number  of  determinations  made  at  the  same 
place  in  different  years,  it  is  not  indispensable  that  there  should 
be  observations  at  both  solstices  in  every  year,  provided  there 
are  in  all  as  many  determinations  at  the  northern  as  at  the 
southern  solstice. 

419.  EXAMPLE. — Find  the  obliquity  of  the  ecliptic  from  the 


OBLIQUITY    OP   THE    ECLIPTIC. 


663 


following  apparent  declinations  of  the  sun's  centre,  observed  at 
the  Washington  Observatory  by  Professor  COFFIN  and  Lieutenant 
PAGE,  with  the  mural  circle. 


1846. 

D 

1846. 

D 

June  16 

23°  21'  56".02 

December  14 

—  23°  14'  17".26 

"  19 

26  28  .19 

15 

17  33  .82 

"  20 

27  6  .79 

«    16 

20  22  .94 

"  23 

26  39  .92 

«    18 

24  32  .69 

«  27 

20  17  .84 

«    21 

27  20  .43 

22 

27  19  .64 

"    23 

26  49  .82 

"    29 

14  1  .20 

Taking  5*  8m  11*. 2  as  the  longitude  of  Washington  from  Green- 
wich, we  find,  for  apparent  noon  at  Washington,  the  following 
values  of  the  sun's  right  ascension  and  latitude  from  the  Nautical 
Almanac : 


1846. 

A 

ft 

1846. 

A 

P 

June  16 

5*  38™  37'.  13 

+  0".18 

December  14 

17*  26™  52'.  73 

+  0".35 

«     19 

5  51     5.77 

—  0  .19 

15 

17   31     18.43 

,  +  0  .46 

"     20 

5  55   15.44 

—  0  .32 

"        16 

17   35    44.38 

+  0  .57 

"     23 

6     7   44.44 

—  0  .63 

"        18 

17   44    36.91 

+  0.72 

"    27 

6  24  22.00 

—  0  .72 

21 

17   67    66.69 

+  0  .70 

"        22 

18     2    23.39 

+  0  .64 

"        23 

18     6    50.09 

+  0  .50 

«        29 

18   33    28.11 

—  0  .19 

Supposing  no  tables  of  the  reduction  at  hand,  let  us  first 
reduce  the  observations  at  the  summer  solstice  by  the  original 
equation  (696).  Subtracting  ft  from  the  observed  values  of  _£>, 
we  then  have 


If 

log  tan  iy 

log  cosec  A 

log  tan  e 

c 

June  16 

23°  21'  55".84 

9.6355081 

0.0018927 

9.6374008 

23°  27'  23".  61 

«  19 

26  28  .38 

.6370823 

03278 

4101 

25  .22 

"  20 

27  7  .11 

.6373056 

00930 

3986 

23  .23 

«  23 

26  40  .55 

.6371524 

02478 

4002 

23  .51 

"  27 

20  18  .56 

.6349452 

24592 

4044 

24  .25 

Apparent  obliquity  =  23    27  23  .96 


664 


OBLIQUITY    OF    THE    ECLIPTIC, 


For  the  sake  of  comparison,  I  add  the  results  of  the  computa- 
tion by  the  series  (697),  which,  however,  will  be  far  less  con- 
venient than  the  above  direct  computation,  unless  a  table  of  the 
reduction  is  used. 


« 

D 

Bed.  to  solstice. 

Fed.  for 
Q  lat. 

June  16 

_|_27m22».87 

23°  21'  56".02 

-1-  5'  27".77 

—  0".18 

»     19 

+    8    54.23 

26  28  .19 

0  56  .84 

+  0  .19 

"     20 

+    4    44.56 

27     6  .79 

0  16  .13 

+  0  .32 

«     23 

—    7    44.44 

26  39  .92 

0  42  .96 

+  0  .63 

"    27 

—  24   22.00 

20  17  .84 

7     5  .69 

+  0.72 

6 

23° 

27'  23 

'.61 

25 

.22 

23 

.24 

23 

.51 

24 

.25 

Apparent  obliquity  =  23  27  23  .96 

Nutation*  +     8  .24 

Reduction  to  Jan.  0.  1846  =  0".4645  X  0.469  =  __  +    0  .22 

Mean  obliquity  1846.0  ==  23  27  32  .42 

In  the  same  manner,  for  the  southern  solstice  we  have : 


u 

D 

Red.  to  solstice. 

Red.  for 
©  lat, 

e 

Dec.  14 

+  33™   7'.27 

—  23°  14'  17".  26 

—  13'     6".48 

—  0".35 

23°  27'  24".09 

"     15 

+  28    41.57 

17  33  .82 

9  50  .24 

—  0.  46 

24  .52 

"     16 

+  24    15.62 

20  22  .94 

7     1  .98 

—  0  .57 

25  .49 

"     18 

+  15   23.09 

24  32  .69 

2  49  .70 

-0.  72 

23  .11 

«     21 

+    2     3.31 

27  20  .43 

0     3  .03 

—  0  .70 

24  .16 

"     22 

—    2    23.39 

27  19  .64 

0     4  .09 

-0  .64 

24  .37 

"     23 

—    6    50.09 

26  49  .82 

0  33  .49 

—  0.50 

23  .81 

"    29 

—  33    28.11 

14    1  .20 

13  23  .05 
Apparent  o' 
Nutation 

+  0.19 
jliquity  = 

24  .06 

23    27  24  .20 

+     8  .98 

Reduction  to  Jan.  0.  1846  =  0".4645  X  0-971  =* 
Mean  obliquity  1846.0  — 

+     0  .45 

23    27  33  .63 

The  results  from  the  two  solstices  being  combined  in  order  to 


*  The  nutation  for  1846  is  found  by  the  formula  (Art.  381) 

As  =  9".2235  cos  &  —  0".0897  cos  2&  +  0".0886  cos  2<£ 

+  0  .5509  cos  20  +0  .0093  cos  (Q  +  r) 

For  the  northern  solstice  June  21,  9*,  I  have  taken  &  ==  214°  27',  <£  =  69°,  Q  =  90°, 
r  =  280° ;  for  the  southern  solstice,  Dec.  21. 16A,  &  =  204°  45',  <[  =  319°,  Q  =  270°, 
r  —  280°.  To  proceed  with  theoretical  rigor,  the  nutation  should  be  found  for  the 
time  of  each  observation. 


EQUINOCTIAL    POINTS.  665 

eliminate  the  error  of  the  assumed  latitude  of  Washington,* 
we  have,  finally, 

Mean  obliquity  for  1846.0  from  observation  =  23°  27'  33".03 
The  same  by  PETERS'S  formula  (646)  with  ) 

»  V   (t  U        09      OK 

the  annual  decrease  0".4645  J  * 

420.  The  secular  variation  of  the  obliquity  is  found  hy  com- 
paring its  values  at  very  distant  epochs.     The  observations  of 
BRADLEY  from  1753  to  1760  gave  for  1757.295  the  mean  obliquity 
23°  28'  14". 055.     The  observations  at  the  Dorpat  Observatory 
gave  for  1825.0  the  mean  obliquity  23°  27'  42".607.     Hence 

o-\ tf  440 

Annual  var.  =  —  -       —  =  —  0".4645 
67.705 

BESSEL  found  —  0".457  by  comparing  BRADLEY'S  observations 
with  his  own. 

The  secular  variation  is  also  found  in  Physical  Astronomy, 
theoretically.  The  value  thus  obtained  by  PETERS  in  his  Nume- 
rus  Constans  Nutationis  is  —  0".4738,  as  given  in  the  formulae 
(646). 

421.  Determination  of  the  equinoctial  points,  and  the  absolute  right 
ascension  and  declination  of  the  fixed  stars. — The  declinations  of  the 
fixed  stars  are  either  directly  measured  by  the  fixed  instruments 
of  the  observatory,  or  deduced  immediately  from  their  observed 
meridian  zenith  distances  (corrected  for  refraction)  by  the  formula 
d  =  <p  —  f .     The  practical  details,  which  depend  on  the  instru- 
ment employed,  will  be  given  in  Vol.  II.     Here  we  have  only 
to  observe  that  the  immediate  result  of  such  a  measurement  is 
the  apparent  declination  at  the  time  of  observation,  which  must 
then  be  reduced  to  the  mean  declination  for  some   assumed 
epoch  by  the  formulae  of  the  preceding  chapter. 

The  position  of  the  equinoctial  points  is  determined  as  soon 
as  we  have  found  the  right  ascension  of  one  fixed  star ;  and  this 
is  done  by  deducing  from  observation  the  difference  between  the 

*  The  latitude  employed  in  deducing  the  declinations  was  38°  53'  39".25.  The 
latitude  given  by  the  culminations  of  Polaris  is  38°  53' 39".  52  (Washington  Astr. 
Obs.,  Vol.  I.,  App.  p.  113).  If  we  adopt  the  latter  value,  the  obliquity  derived  from 
the  northern  solstice  will  be  increased  by  0".27,  and  that  derived  from  the  southern 
solstice  will  be  diminished  by  the  same  quantity ;  and  the  difference  then  remaining 
between  the  two  results  will  be  only  0".67. 


606  EQUINOCTIAL    POINTS. 

sun's  right  ascension  and  that  of  the  star  at  the  time  the  sun  is 
at  the  equinoctial  points.  For  this  purpose  a  bright  star  is 
selected,  which  can  be  observed  in  the  daytime  and  at  either 
equinox,  and  which  is  not  far  from  the  equator.  On  a  day  near 
the  equinox  the  times  of  transit  of  the  sun  and  the  star  are 
noted  by  the  sidereal  clock ;  and  at  the  time  of  the  sun's  transit 
his  declination  is  also  measured.  Let 

T  =  the  clock  time  of  the  sun's  transit, 
t  =          "  "         "        star's      " 

Aj  D,  fi  =  the  sun's  apparent  right  ascension,  declination, 

and  latitude  at  the  time  T, 

a  =  the  star's  apparent  right  ascension  at  the  time  t, 
e  =  the   apparent   obliquity   of   the   ecliptic   at   the 

time  T; 

then,  correcting  the  sun's  declination  by  the  formula  (695),  or, 

D'  =  D  —  p  sec  s  cos  D 
we  have,  by  (694), 

sin  A  =  tan  D'  cot  s  (699) 

Thus  A  becomes  known,  and  hence,  also,  a  by  the  formula 

a  =  A  +  (t  —  T)  (700) 

in  which  t  —  T  is  the  true  sidereal  interval  between  the  observa- 
tions corrected  for  the  clock  rate. 

The  observation  is  to  be  repeated  on  a  number  of  days  pre- 
ceding and  following  each  equinox.  The  star's  apparent  right 
ascension  is  in  each  case  to  be  freed  from  the  effects  of  aberra- 
tion, nutation,  and  precession  (also  proper  motion  and  annual 
parallax,  if  known).  Each  observation  thus  furnishes  a  value  of 
the  star's  mean  right  ascension  at  the  epoch  to  which  the  re- 
duction is  made.  In  order  to  learn  what  combination  of  these 
values  will  best  eliminate  constant  errors  in  the  elements  upon 
which  A  depends,  let  us  examine  the  effects  of  these  errors. 
"We  speak  only  of  constant  errors ;  the  accidental  errors  of  obser- 
vation being  reduced  to  their  minimum  effect  by  taking  the 
mean  of  a  large  number  of  observations. 

The  correction  which  the  assumed  value  of  the  obliquity 
requires  being  denoted  by  de,  the  corresponding  correction  of  A 
is  found,  by  differentiating  (699),  to  be 


EQUINOCTIAL  POINTS.  667 

2  tan  A 


dA=—ds 


sin 


The  correction  of  the  declination  D'  is  composed  of  the  cor- 
rections in  the  latitude  ^>,  and  the  zenith  distance  £ ;  since,  by 
the  formula  D  =  <p  —  £,  we  have 

dD  =  d<p  —  d: 

But  d£  is  itself  composed  of  the  corrections  required  in  the  re- 
fraction and  the  sun's  parallax  and  the  correction  for  any  error 
peculiar  to  the  zenith  distance  f ,  which  affects  the  meridian  in- 
strument employed  in  the  observation.  Denoting  the  correction 
of  the  refraction  by  dr,  that  of  the  sun's  parallax  by  dp  sin  f , 
that  of  the  instrument  for  the  zenith  distance  £  by/(£),  we  have 

dD  =  d<p  —  \_dr  —  dp  sin  C  +  /(C)] 

The  effect  of  this  correction  upon  A  is  found,  by  differentiating 
(699)  with  reference  to  D'  (regarding  dD  as  equal  to  dDf],  to  be 

2  tanJ. 
dA  =  dD  - 

sin  27)' 

If  then  a/  denotes  the  corrected  mean  right  ascension  of  the 
star,  free  from  all  constant  errors,  we  have 

r  -.  2  tan  A        7   2  tan  A 

a'=  a  +  [df  —  dr  .+  dp  sin  C  -./C)]  -— —  -  dt  — 

J  sin  21) '  sin  2e 

This  formula  shows  that  nearly  all  the  errors  will  be  eliminated 
by  taking  the  mean  between  two  observations  taken  at  the  same 
zenith  distance  (or  the  same  declination),  the  one  near  the  vernal, 
the  other  near  the  autumnal  equinox.  For,  the  first  observation 
being  taken  when  the  declination  is  D'  and  right  ascension  A, 
at  the  second  one  the  same  declination  Df  will  give  the  right 
ascension  180°  —  A,  the  tangent  of  which  is  the  negative  of  that 
of  A.  The  temperature  being  generally  different  at  the  two 
seasons  of  the  year,  we  cannot  assume  that  the  error  in  the 
refraction  tables  will  be  the  same  at  both  observations  unless  we 
can  also  assume  that  the  law  of  correction  of  the  refraction  for 
temperature  is  perfectly  known.  So,  also,  we  must  admit  the 
possibility  that  such  changes  of  temperature  change  the  instru- 
mental correction ;  but  the  corrections  of  the  latitude  and  the 
parallax  will  remain  the  same.  Hence,  if  ax  is  the  mean  right 


668  EQUINOCTIAL    POINTS. 

ascension    computed    from    the   observation   at  the   autumnal 
equinox,  the  corrected  right  ascension  will  be 

-,  2  tan  A  2  tan  A 


in  which  drl  and/,(£)  denote  the  corrections  for  the  same  zenith 
distance  as  before,  but  for  a  different  temperature.  The  mean 
value  of  a'  obtained  from  the  two  observations  is  then 

a'  =  J  (.  +  at)  +  [dr,  -  dr  +  /,  (0  -/(O]  L 


This  mean  is  thus  freed  entirely  from  the  effects  of  the  errors  of 
latitude  and  the  assumed  obliquity,  and  the  remaining  error  is  com- 
posed merely  of  the  difference  of  the  errors  of  refraction  and  of  the 
instrument  arising  from  differences  of  temperature.  The  differ- 
ence of  temperature  at  the  vernal  and  autumnal  equinoxes, 
though  considerable,  is  not  so  great  but  that  we  may  assume  the 
quantity  drl  —  dr  to  be  evanescent  in  the  present  state  of  the 
refraction  tables.  To  eliminate  the  effects  of  temperature  upon 
the  instrument,  the  only  course  is  to  make  a  special  investigation 
of  its  errors  at  various  temperatures. 

It  follows  from  this  discussion  that  the  absolute  right  ascen- 
sion of  a  star  can  be  accurately  determined  by  means  of  observa- 
tions at  both  equinoxes  so  arranged  that  for  every  observation 
near  the  vernal  equinox  at  the  right  ascension  A  there  will  be  a 
corresponding  one  at  the  autumnal  equinox  at  the  right  ascension 
180°  —  A.  This  condition  is  satisfied  nearly  enough  by  regarding 
as  corresponding  observations  those  which  are  taken  between 
the  declinations  0°  and  -f  2°  after  the  vernal  and  before  the 
autumnal  equinox,  between  0°  and  —  2°  before  the  vernal  and 
after  the  autumnal  ;  between  +  2°  and  +  4°  ;  —  2°  and  —  4°,  &c. 
On  account  of  the  very  complete  elimination  of  errors,  it  is  safe 
to  extend  the  observations  even  as  far  as  -f  14°  and  —  14°.* 

EXAMPLE.  —  The  following  observations  of  the  sun  and  f  Pcgasi 
on  the  meridian  were  taken  at  the  "Washington  Observatory  in 
the  year  1846  :f 

*  BESSEL  :  Fundamenta  Astronomies,  pp.  12,  14. 

f  The  transits  were  taken  with  the  "West  Transit,"  the  declinations  with  the 
Mural  Circle.  Both  the  first  and  second  limbs  of  the  sun  were  observed  on  the  seven 
threads  of  the  transit  instrument,  and  the  declination  of  both  the  north  and  the  south 
limbs  with  the  mural. 


EQUINOCTIAL    POINTS.  669 


Feb.  23.  D  =  —  9°  46'  15".85 
a  a  T  =  22*  ^6— 28*11 
"  "  t  =  0  5  18 .99 


Oct.  17.  D  =  —  9°  17'  53".12 
«  "  T=  13*  28-40-.01 
"  16.  t  =  05  22 .97 


The  times  of  transit  are  corrected  for  the  supposed  error  and 
rate  of  the  clock. 

For  the  dates  of  the  two  observations,  the  apparent  obliquity 
of  the  ecliptic  and  the  sun's  latitude  are  as  follows  : 

Feb.  23.  Oct.  17. 

e        23°  27'  26".10  23°  27'  24".35 

0               +     0  .33  0  .13 
whence 

-  /?  sec  e  cos  D                     0  .35  +0  .14 

D'-     9    46  16  .20  -  9    17  52  .98 

log  tan  D'     7i9.236063  7i9  214105 

log  cot  e         0.362585  0.362595 

log  sin  A     7i9.598648  7i9.576700 

A        22*  26-  28M7  13*  28-  40M4 

A  —  T              -f      0.06  +0.13 

t  -f  A  —  T  =  a         0     5    19 .05  0     5    23 .10 

Keduction  to  1850.0               -f    12 .15  +       8 .12 

Mean  a  for     1850.0  =    0      5    31 .20  0     5    31 .22 

The  reduction  to  1850  is  here  used  because  it  can  be  taken 
directly  from  the  general  tables  for  reducing  the  apparent  places 
of  stars  to  mean  places,  given  in  the  volume  of  Washington 
Observations  for  1847.  Taking  the  mean  of  the  two  observations, 
we  have,  finally, 

Mean  E.  A.  of  y  Pegasi  for  1850.0  =  0*  5-  31-.21 

422.  When,  by  the  combination  of  a  great  number  of  observa- 
tions, the  right  ascension  of  a  fundamental  star  is  thus  established, 
the  right  ascensions  of  all  other  stars  follow  from  the  differences 
of  time  between  their  several  transits  and  that  of  the  fundamental 
star.  But,  in  the  present  state  of  the  star  catalogues,  it  will  be 
preferable  not  to  limit  the  object  of  these  observations  to  deter- 
mining a  single  star.  The  constant  use  of  the  same  fundamental 
stars  as  "  clock  stars"  (stars  near  the  equator  by  which  the  clock 
correction  and  rate  are  found)  gives  to  the  relative  right  ascensions 
of  these  stars  (as  derived  from  all  their  observed  transits  during 
one  or  more  years)  a  high  degree  of  accuracy.  Assuming, 


670  EQUINOCTIAL    POINTS. 

therefore,  that  the  relative  right  ascensions  of  the  clock  stars  are 
correct,  the  object  of  our  observations  of  the  sun  will  'be  to 
determine  the  common  correction  of  the  absolute  right  ascensions 
of  all  these  stars.  Accordingly,  if  we  deduce  the  sun's  apparent 
right  ascension  directly  from  each  observation  by  applying  to  the 
clock  time  of  the  transit  of  the  sun's  centre  the  clock  correction 
obtained  from  the  fundamental  stars,  and  compare  this  with  the 
apparent  right  ascension  computed  from  the  observed  declination, 
we  have  the  correction  which  the  right  ascensions  of  these  stars 
require.  All  that  has  been  said  above  respecting  the  grouping 
of  the  observations  at  the  two  equinoxes,  of  course,  applies 
equally  well  to  this  process. 

Thus,  in  the  preceding  example,  taking  the  clock  times  of  the 
sun's  transits  there  given  as  the  directly  observed  right  ascensions 
(since  they  have  actually  been  corrected  for  the  clock  error 
obtained  from  a  number  of  fundamental  stars),  we  shall  have 

Feb.  23.  Oct.  17. 

Observed  R.  A.  of  0,          22*  26-  28-.11  13*  28"*  40-.01 

Computed    "        "              "     "    28 .17  "     "    40.14 

Correction  of  clock  stars,        -f     0 .06  -f-     0 .13 

whence 

Mean  correction  of  the  E.  A.  of  the  clock  stars  =  -f-  O'.IO 


CONSTANT   OF   REFRACTION.  671 


CHAPTER  XIII. 

DETERMINATION   OF   ASTRONOMICAL   CONSTANTS   BY  OBSERVATION. 

423.  I  SHALL  not  attempt  to  enter  into  all  the  details  of  the 
methods  by  which  the  various  astronomical  constants  are  deter- 
mined from  observations,  but  shall  confine  myself  to  a  sketch  of 
their  general  principles,  which  will  serve  as  an  introduction  to 
the  special  papers  to  be  found  in  astronomical  memoirs  and 
other  sources. 

THE   CONSTANTS   OF   REFRACTION. 

424.  The  general  refraction  formula  (191)  involves  the  two 
constants  a  and  /9,  both  of  which  may  be  found  from  theory  by 
the  formulae  (ITS)  and  (176).    But,  as  the  refraction  formula  was 
deduced  from  an  hypothesis,  it  was  not  to  be  expected  that  the 
theoretical  values  of  a  and  /9  would  give  refractions  in  entire 
accordance  with  observation.     The  discrepancies,  however,  are 
exceedingly  small  :  so  small,  indeed,  that  the  formula  may  be  re- 
garded as  representing  well  enough  the  law  of  refraction,  with- 
out resorting  to  any  new  hypothesis  ;  and  to  perfect  it  we  have 
only  to  give  the  constants  slightly  amended  values,  whereby  the 
computed  refractions  are  made  to  harmonize  entirely  with  those 
deduced  from  observation.    To  deduce  the  corrections  of  a  and  /?, 
we  can  employ  the  concise  expression  of  the  refraction  (213),  or 


The  factor  1  —  a  differs  so  little  from  unity  that  we  may  regard 
it  as  constant  in  determining  the  small  correction  of  r,  and, 
therefore,  by  differentiating,  we  have 


(l-a)rfr=sm'^|^.<fa+(^-^ 


By  (217)  and  (210)  we  have 

dQ      dQ   dx       1—  x 


da        dx    da,  x       sin2  z 


672  CONSTANT   OF   REFRACTION. 

where  Q'  is  known  by  (218),  and,  therefore,  the  coefficient  of 
da  can  be  computed.  Also,  since  —  is  given  by  (220),  and  Q  by 

(212),  the  coefficient  of  dp  is  known.  We  should,  however,  find 
the  correction  of  the  constant  a0,  or  that  which  corresponds  to 
the  normal  temperature  and  barometric  pressure  of  the  refrac- 
tion table.  By  (205)  we  have 

da0  p 


As  for  d/3,  it  is  evidently  the  same  as  dj30. 

But,  since  a0  can  require  but  a  very  small  correction,  great  pre- 
cision in  the  coefficient  of  da0  is  not  necessary;  and,  if  we 
neglect  the  second  and  higher  powers  of  a0,  it  is  easily  seen  that 

this  coefficient  will  be  reduced  to  — ,  r  being  the  refraction  com- 
puted for  the  actual  state  of  the  air  by  the  tables.  This  amounts 
to  assuming  that  r  and  dr  vary  directly  in  proportion  to  a0  and  etot0; 
an  assumption  which  is  very  nearly  correct,  as  can  be  seen  from 
the  approximate  formula  (159),  in  which  we  have  very  nearly 
2kdQ  =  a0.  We  may  also  in  our  differential  formula  put  unity 
in  the  place  of  the  factor  1  —  a ;  and  hence  if  we  put 


A----- 


d-V--Q\ 
4ft        2/9  / 


we  shall  have 

dr  =  Ada0+Bdp0  (701) 

It  only  remains  to  show  how  this  differential  forrnula  is  to  be 
applied  in  deducing  da0  and  dft0.  The  observations  best  suited 
to  our  purpose  are  those  of  the  zenith  distance  of  a  circumpolar 
star  at  its  upper  and  lower  culminations.  Let 

zf,  2j'  =  the  observed  zenith  distances  above  and  below  the 

pole  respectively, 
z,  zl  =  the  true  zenith  distances  obtained  by  employing  the 

tabular  refraction, 
d,  Sl  —  the  declination  of  the  star  at  the  two  culminations 

respectively, 
<p  =  the  assumed  latitude  of  the  place  of  observation. 


CONSTANT  OF  SOLAR  PARALLAX.  673 

The  true  zenith  distances  which  would  be  obtained  by  a  table 
of  refractions  founded  on  the  corrected  constants  will  be  z  -f  dr 
and  zl  -f-  drv  ;  and,  therefore,  if  dtp  denotes  the  correction  of  the 
assumed  latitude,  we  shall  have 

90°  —  (<p  -f  d<p)  =  z  -f  dr  +   90°  —  8 
90°  —(*+  dj)  =  z,  +  dr,—  (90°  -  ax) 

whence,  by  taking  the  mean, 

90°  -  <p  -  d?  =  J  (z  +  z,)  +  £('\  -  *)  +  Kdr  +  dr,) 

The  quantity  ^  —  <?  is  merely  the  very  small  change  of  the 
star's  declination  between  the  two  culminations,  arising  from 
precession  and  nutation,  which  is  accurately  known.  If  we  sub- 
stitute the  values  of  dr  and  drl  in  terms  of  da  and  dp,  and  then 
put 


n  =  K*  +  *,)  +  K*i  —  *)  +  ?  -  90° 
we  have  the  equation  of  condition 

?0  -f  n  =  0  (702) 


By  employing  a  number  of  stars  which  culminate  at  various 
zenith  distances,  we  shall  obtain  a  number  of  such  equations,  in 
which  the  coefficients  a  and  b  will  have  different  values  :  so  that 
the  solution  of  all  these  equations  by  the  method  of  least 
squares  will  determine  the  three  unknown  quantities  efy,  da0, 
and  df)Q. 

THE  CONSTANT  OF  SOLAR  PARALLAX. 

425.  The  constant  of  solar  parallax  is  the  sun's  mean  equatorial 
horizontal  parallax^  or  its  horizontal  parallax  when  its  distance 
from  the  earth  is  equal  to  the  semi-major  axis  of  the  earth's 
orbit.  The  constant  of  parallax  of  any  planet  is  also  its  parallax 
when  its  distance  from  the  earth  is  equal  to  the  semi-major  axis 
of  the  earth's  orbit:  so  that  the  constant  of  solar  parallax 
belongs  to  the  whole  solar  system. 

The  relative  dimensions  of  the  orbits  of  the  planets  are  known 
from  the  periodic  times  of  their  revolutions  about  the  sun,  since, 
by  KEPLER'S  third  law,  the  squares  of  their  periodic  times  are 
proportional  to  the  cubes  of  their  mean  distances  from  the  sun, 

VOL.  I.—  43 


674  CONSTANT    OF    SOLAR    PARALLAX. 

that  is,  to  the  cubes  of  the  semi-major  axes  of  their  orbits.  The 
ratios  of  these  distances  are  therefore  known. 

Again,  the  form  and  position  of  each  orbit  are  known  from 
Physical  Astronomy  ;*  and  therefore  the  ratio  of  the  planet's 
distance  from  the  earth  at  any  given  time  to  the  earth's  mean 
distance  is  also  known. 

According  to  these  principles,  if  the  distance  of  any  planet 
from  the  earth  can  be  found  at  any  time,  the  dimensions  of  all 
the  orbits  are  also  found :  in  other  words,  when  we  have  found 
the  parallax  of  one  planet  we  have  also  found  that  of  all  the 
planets,  as  well  as  that  of  the  sun. 

426.  To  find  a  planets,  or  the  sun's,  parallax  by  meridian  observa- 
tions.— Let  the  meridian  zenith  distance  of  the  planet's  centre 
be  observed  on  the  same  day  at  two  places  nearly  on  the  same 
meridian,  but  in  very  different  latitudes.  After  correcting  the 
observed  quantities  for  refraction,  let 

C',  C/  =  the  apparent  zenith  distances  at  the  north  and  south 

places  of  observation,  respectively, 
C,   d  =  the  true  (geocentric)  zenith  distances, 
p,  pl  =  the  parallax  for  the  zenith  distances  C  and  d» 
TT,  TT,  =  the  equatorial  horizontal  parallax  at  the  respective 

times  of  observation, 

J,  Jj  =  the  geocentric  distances  of  the  planet  at  these  times. 
d,  dv  =  the  geocentric  declination  of  the  planet  at  the  same 

times, 

TTO  =  the  sun's  mean  equatorial  horizontal  parallax, 
J0  =    "      "          "       distance  from  the  earth, 
R  =  the  earth's  equatorial  radius. 

Also  for  the  places  of  observation  let 

y>,  <f>l  =  the  astronomical  latitudes, 

y>',y>i=  the  reduced  or  geocentric  latitudes, 

pj  p1  =  the  radii  of  the  terrestrial  spheroid  for  these  latitudes. 


We  have 


R  R  .  R 

sin  TT  =  — -  sin  rr.  =  - —  sin  xn  =  — • 


*  They  are  found  from  three  complete  observations  of  the  right  ascension  and 
declination  of  each  planet  at  three  different,  times  (GAUSS,  Theoria  Motus  Corporum 
CiElfistium),  and  therefore  from  the  observed  directions  of  the  planet,  the  absolute 
distance  being  unknown. 


CONSTANT    OF    SOLAR    PARALLAX.  675 

and  therefore 


sin  -K  =  —  -  sin  TT  sin  7    =  —  5 


The  quantities  A  and  Al  are  to  be  found  from  the  planetary 
tables,  or  directly  from  the  Nautical  Almanac,  where  they  are 

expressed  in  terms  of  J0  as  the  unit:  so  that  their  values  there 

j  J 

given  are  the  values  of  the  ratios  —  and  —  .     Hence  we  shall  put 

0      *  4) 

J0  =  1  in  the  preceding  formulae,  and  also  put  the  arcs  for  their 
sines  (since  the  greatest  planetary  parallax  is  only  35")  :  so  that 
we  have 


Then,  by  (114), 

p  =  p  n  sin  [r'  —  (p  —  <p'y]  =  ^-°  sin  [C'  —  (p  —  /)] 

J^i :==:  /'i^i  ^^  C*/  —  C^i — ^/)]  ""^  — ~~  ®^  L^/  —  (vi — ^i')l 

i 

But  we  also  have 
and  hence 

from  which  we  obtain 

As  the  small  difference  3  —  3l  will  be  accurately  known,  the 
observations  being  taken  nearly  on  the  same  meridian,  all  the 
quantities  in  the  second  member  of  this  equation  may  be 
regarded  as  known.  Hence,  putting 

n  =  £'  —  c/ (<p yj  +  (<*  —  £,) 

(703) 

we  obtain  the  equation 

axa=n  (704) 

which  determines  XQ.  If  the  zeniths  of  the  two  places  of  obser- 
vation are  on  opposite  sides  of  the  star  (which  is  the  most  favor- 


676  CONSTANT    OF    SOLAR    PARALLAX. 

able  case),  the  zenith  distance  at  the  southern  place  must  be 
taken  with  the  negative  sign  in  the  above  formulae.  The  coeffi- 
cient a  then  becomes  an  arithmetical  sum,  and  it  is  evident  that 
the  greater  the  value  of  a,  the  greater  will  be  the  degree  of  accu- 
racy in  the  determination  of  TTO. 

But,  in  order  to  give  this  method  all  the  precision  necessary 
in  finding  so  small  a  quantity  as  TTO,  the  quantity  n  must  not 
depend  upon  the  absolute  zenith  distances  observed  (which 
involve  the  errors  of  divided  circles  and  the  whole  errors  of  the 
refraction  table  at  these  zenith  distances),  nor  upon  the  quantity 
9  —  ^i  (which  involves  the  errors  in  the  latitudes  of  the  places), 
but  upon  micrometric  measures.  For  this  purpose,  the  planet  is 
compared  with  a  star  nearly  in  the  same  parallel  of  declination, 
and  always  with  the  same  star  at  both  places  of  observation,  the  com- 
parison stars  being  previously  selected  and  agreed  upon  by  the 
observers.  The  star  and  planet  should  differ  so  little  in  declina- 
tion that  they  will  both  pass  through  the  field  of  the  meridian 
telescope,  the  instrument  remaining  firmly  clamped  between  the 
transits  of  the  two  objects ;  and  then  the  difference  of  apparent 
decimation  of  the  planet  and  star  will  be  directly  measured  with 
the  micrometer.  This  difference  is  to  be  corrected  for  the  dif- 
ference of  refraction  at  the  zenith  distances  of  the  planet  and 
star,  which  difference  of  refraction,  being  very  small,  can  be 
computed  with  the  greatest  accuracy.*  If  then 

D  —  the  declination  of  the  star, 

A<5,  A^  =  the  observed  differences  of  declination  of  the  star 
and  planet  (corrected  for  refraction)  at  the  two 
places  of  observation, 

the  observed  apparent  declination  of  the  planet  at  the  northern 

place  is 

D  _j_  A,j  =  ^  _  £' 

and  at  the  southern  place 

D+A,5i  =  ¥>i_Cl' 
whence 

Aa  _  A,5i=  _  c'  -  c/)  +  (?  -  ?1) 

and  the  value  of  n  in  (703)  becomes 

n  =  3  —  dl  —  (A.5  —  A^)  (705) 

*  Vol.  II.    Correction  of  micrometer  observations  for  refraction. 


CONSTANT    OF    SOLAR    PARALLAX.  677 

where  A£  and  A<5t  are  in  each  case  the  planet's  declination  minus 
the  star's  declination,  and  their  signs  are  to  be  carefully  observed. 
For  computing  the  coefficient  a,  the  apparent  zenith  distances 
will  be  obtained  by  the  formulae 

C'  =  ?  —  (D  +  A(5)  C/  =  9l  —  (D  +  A.\) 

so  that  we  have 

a==£  sin  [>'  -  (D  +  A*)]  -  £  sin  [>/  -  (D  +  A^)]    (706) 

and  then,  as  before, 

#  7r0=  n 

A  great  number  of  such  corresponding  observations  will  be 
necessary  in  order  to  determine  TTO  with  accuracy;  and  all  the 
equations  of  the  form  just  given  are  to  be  combined  by  the 
method  of  least  squares.  Thus,  from  the  equations 


j  a"7r0  =  n",  &c. 

we  obtain  the  final  equation 

[aa]  r0  =  [an]  or  TTO  =  i~l 

in  which  [aa]  =  aa  -\-  a'  a'  +  a"  a"  +  &c-5  an(i   [«w]  =  an  +  a'/i' 
+  «'  'n"  +  &c. 

427.  Jb  find  ike  solar  parallax  "by  extra-meridian  observations  of  a 
planet.  —  The  preceding  process  will  require  but  a  slight  modifi- 
cation. The  difference  of  apparent  declination  of  the  planet  and 
a  neighboring  star  is  measured  at  both  stations  with  a  micrometer 
attached  to  an  equatorial  telescope,  and  is  to  be  corrected  for 
refraction.  The  quantity  n  will  then  be  found  by  (705).  The 
coefficient  a  will  now  be  the  difference  of  the  coefficients  of 
parallax  in  declination,  computed  by  the  formulae  (143),  accord- 
ing to  which,  if  we  put 

tan  <p'  tan  a' 

tan  f  =.-  -  tan  y.  =  - 

COS  (0  —  a)  COS(01—  aj 


678  CONSTANT    OF    SOLAR    PARALLAX. 

we  shall  have 

a=  P   sin  /  sin  fr  —  3)        Pl  sin  ?/  sin  fc  —  ^) 
J  sin  f  Jt  sin  ^ 

in  which  0  and  0X  are  the  local  sidereal  times  of  the  observations, 
a  and  ax  the  right  ascensions,  o  and  ^  the  declinations  of  the 
planet  at  these  times.  The  equation  of  condition  from  each  pair 
of  corresponding  observations  of  the  same  star  will  then  be,  as 
before,  a  TTO  =  n. 

If  several  comparisons  are  made  at  either  place  on  the  same 
day,  these  must  first  be  combined,  and  reduced,  as  it  were,  to  a 
single  comparison.  Thus,  if  we  put 

p    sin  <'  mi\  (    —  <5 


A  sin  f 

we  have,  for  each  comparison  of  the  planet  with  the  star, 

8  =  D  -f  A<5  -f  CTTO 
and  if  m  such  comparisons  are  made,  their  mean  will  be 


In  like  manner,  at  the  second  place,  we  shall  have  for  ml  obser- 
vations the  equation 


and,  taking  the  difference  of  these  equations,  we  shall  put 

/Z(A<5)          ^(A'V)\ 

1      \     m  mi     t 


m 


The  equation  of  condition  a  TTO  =  n  will  then  represent  all  the 
observations  on  the  same  day  at  the  two  places. 

428.  The  equations  of  condition  will  involve  smaller  numbers 
and  be  more  easily  solved  if  the  unknown  quantity  is,  not  the 
whole  parallax,  but  the  correction  of  some  assumed  value  of  the 
parallax  not  greatly  in  error.  In  this  case  we  may  correct  each 
observed  difference  A£  for  parallax,  employing  the  assumed  value 


CONSTANT   OF   SOLAR   PARALLAX.  679 

of  TTO;  and,  proceeding  as  before,  we  shall  have  the  equation  of 
condition  a  ATTO  =  w,  in  which  A/TO  is  the  required  correction  of  ~0. 

429.  If  but  one  limb  of  the  planet  is  observed  at  one  or  both 
the  stations,  it  will  be  necessary  to  introduce  the  correction  for 
the  semidiameter.     As  the  semidiameter  itself  should  then  be 
regarded  as  an  unknown  quantity,  to  be  found  if  possible  from 
the  observations,  its  complete  expression,  in  terms  of  all  the  cor- 
rections which  the  observations  may  require,  is  to  be  employed. 
This  will  be  found  in  Article  435. 

430.  The  differences  of  right  ascension  of  the  planet  and  a 
neighboring  star  may  also  be  employed  in  the  same  manner  as 
the  differences  of  declination,  the  places  of  observation  being  in 
that  case  in  widely  different  longitudes.     We  have  only  to  intro- 
duce into  (707)  the  coefficients  of  the  parallax  in  right  ascension 
computed  by  the  first  equation  of  (143),  and  in  the  expression  of 
n  substitute  right  ascensions  for  declinations. 

431.  The  only  planets  which  are  near  enough  to  the  earth  for 
the  successful  application  of  this  method  are  Mars  and  Venus. 

Mars  is  nearest  to  the  earth  at  the  time  of  opposition,  and  for 
this  time  the  British  Nautical  Almanac  furnishes  an  Ephemeris 
of  stars  to  be  observed  with  the  planet.  All  the  oppositions, 
however,  are  not  equally  favorable.  The  mean  distance  of  Mars 
from  the  sun  being  —  1.524,  and  the  eccentricity  of  the  orbit 
—  0.0933,  while  the  mean  distance  of  the  earth  =  1  and  the  ec- 
centricity of  its  orbit  =  0.017,  it  follows  that  for  an  opposition 
in  which  Mars  is  at  its  perihelion  while  the  earth  is  at  its 
aphelion,  the  distance  of  the  two  bodies  will  be  0.365 ;  but  for 
one  in  which  Mars  is  at  its  aphelion  and  the  earth  at  its  peri- 
helion, their  distance  will  be  0.683.  Thus  the  former  case  will 
be  nearly  twice  as  favorable  as  the  latter. 

Venus  is  nearest  to  the  earth  at  the  time  of  inferior  conjunc- 
tion, but  at  that  time  can  very  rarely  be  compared  micrometric- 
ally  with  stars,  as  the  observations  would  be  made  with  the  sun 
above  the  horizon.  The  most  favorable  position  of  this  planet 
is  at  or  near  its  stationary  points,  where  the  changes  of  the 
planet's  place  are  small  and  may  therefore  be  accurately  com- 
puted, while  the  distance  from  the  earth  is  still  not  too  great.* 

*  GERLING,  Astron.  Nach.,  No.  599. 


680  CONSTANT    OF    LUNAR    PARALLAX. 

The  United  States  Astronomical  Expedition  to  Chili  under 
Lieut.  J.  M.  GILLISS,  in  the  years  1849-52,  was  set  on  foot  for 
the  purpose  of  determining  the  solar  parallax  by  the  above 
method.  That  indefatigable  and  accurate  observer  collected  a 

o 

large  mass  of  valuable  material,  a  great  part  of  which,  however, 
could  not  be  used  in  the  manner  originally  intended,  for  want 
of  corresponding  observations  at  northern  observatories.  In  the 
thorough  discussion  of  this  material  by  Dr.  B.  A.  GOULD*  will 
be  found  a  full  exposition  of  the  modifications  which  the  method 
required  in  order  to  make  use  of  all  the  observations. 

The  constant  of  solar  parallax  is  also  found  by  the  transits  of 
Venus  over  the  sun's  disc,  Art.  356. 


THE    CONSTANT    OF    LUNAR    PARALLAX. 

432.  The  constant  of  lunar  parallax  is  the  moon's  mean  equatorial 
horizontal  parallax,  or  the  equatorial  horizontal  parallax  corre- 
sponding to  the  moon's  mean  distance  from  the  earth.f 

To  find  the  moon' s  parallax  by  meridian  observations  at  two  stations 
on  the  earth's  surface. 

The  stations  will  be  assumed  to  be  in  opposite  hemispheres  of 
the  earth:  so  that  at  every  observation  the  moon  will  culminate 
south  of  the  zenith  of  the  northern  station,  and  north  of  the 
zenith  of  the  southern  station.  They  will  also  be  assumed  to  be 
nearly  on  the  same  meridian.  At  each  station,  the  apparent 
declinations  of  the  moon's  bright  limb  at  the  instants  of  transit 
are  to  be  observed  on  the  same  day,  and,  consequently,  since  the 
meridians  are  not  remote,  at  nearly  the  same  time.  In  order  to 
eliminate  constant  errors  of  the  refraction  tables  and  instrumental 
errors,  the  difference  of  the  moon's  declination  and  that  of  a  star 
nearly  in  the  same  parallel  is  to  be  observed,  and  the  same  com- 
parison stars  should  be  used  at  both  stations.  The  observed 
difference  of  declination  is  to  be  corrected  for  the  difference  of 
the  refraction  at  the  zenith  distance  of  the  moon  and  star,  and 
then  applied  to  the  assumed  declination  of  the  star.  We  shall 
thus  obtain  the  apparent  declination  of  the  moon's  limb  affected 
only  by  parallax.  Let 

*  U.  S.  Naval  Expedition  to  Chili,  Vol.  III. 

f  The  constant  adopted  in  the  lunar  tables  is  for  the  mean  distance  affected  by  the 
-constant  part  of  the  perturbations  of  the  radius  vector. 


CONSTANT    OF    LUNAR    PARALLAX.  681 

<5,  <5X  =  the  apparent  declinations  of  the  limb  observed  at 

the  north  and  south  stations  respectively, 
D,  Dl  =  the  geocentric  declinations  of  the  moon's  centre  at 

the  respective  times  of  observation, 
<p,  <pl  =  the  geographical  latitudes  of  the  stations, 
f,  YI  —  the  reductions  of  the  latitudes  for  the  earth's  com- 

pression, 
p,  pi  =  the  distances  of  the  stations  from  the  earth's  centre, 

the  equatorial  radius  being  unity, 

P,  Pl  =  the  moon's  horizontal  parallax  at  the  times  of  the 
observation,  respectively; 

then,  the  apparent  zenith  distance  of  the  limb  and  the  geocentric 
zenith  distance  of  the  centre  of  the  moon  being,  for  the  northern 
station, 

C'  =  ^  —  <S  and  C  =  ^  —  -^ 

we  have,  by  (255), 

sin  (D  —  <5)  =  [p  sin  (C'  —  r)  =F  *]  sin  P 

where  k  is  the  constant  ratio  of  the  radii  of  the  moon  and  the 
earth,  for  which  the  value  0.272956  may  be  assumed;  and  the 
upper  or  lower  sign  of  k  is  to  be  used  according  as  the  upper  or 
lower  limb  is  observed. 

At  the  southern  station  we  have 


and  hence,  taking  the  reduction  ft  as  a  positive  quantity, 
sin  (Pi  —  ^)  =  —  fo  sin  (:/  —  ri)  ±  k\  sin  Pt 

where  the  sign  of  k  is  reversed,  since  the  same  limh  will  be  an 
upper  limb  at  one  station  and  a  lower  limb  at  the  other.  For 
brevity,  put 

m  =  p  sin  (C'  —  Y  )  H3  ^ 
m1=/i>1sin(C1  —  ft)  ±k 

then,  from  the  equations 
sin  (D  —  5)  ==  m  sin  P 


582  CONSTANT    OF    LUNAR    PARALLAX. 

we  derive,*  neglecting  powers  of  sin  P  above  the  third, 

m  sin  P       j    m8  sin3  P 
sin  I"         g  '    sin  1" 
33 
~     = 


sin  1"  sin  1" 

If  now  the  times  of"  the  two  observations  reckoned  at  the 
same  first  meridian  are  T  and  JJ,  and  for  the  middle  time 
t  =  J(jT+  7!)  we  deduce  from  the  lunar  tables  the  hourly  in- 

crease of  the  moon's  declination,  or  -,  —  ,  we  shall  have,  with 
regard  to  second  differences, 


Again,  if  we  denote  the  moon's  horizontal  parallax  at  the  time 
t  by  p,  and  compute  from  the  tables  its  hourly  increase  for  this 

time,  or  •—»  we  shall  have 
at 

sin  P  =  sin_p  +  cos  p  sin  I"  (T  —  f)  -| 
sin  P1=  sin;?  -f-  cos^>  sin  l"^  —  *)  -j- 

Taking  the  difference  of  the  above  values  of  D  —  o  and  Di  —  319 
we  obtain,  therefore, 

0  =  [(T,  -  T)  ™    -  ft  -  *)]  sin  1"  +  (m3  +  »»,-)  ^ 
-f  cos  ^  sin  1"     ?  [m  (!T  -  0  +  m,  (T,  -  *)] 


(m  _j_  mi)  8in  |>  (708) 


The  parallax  is  sufficiently  well  known  for  the  accurate  compu- 
tation of  the  terms  in  sin3^  and  y-:  so  that  the  only  unknown 
quantity  in  this  equation  is  the  last  term.  In  this  term  we  have 

m  +  ml  =  p  sin  (:'  -  r}  +  Pl  sin  (:/  -  Tl)  (709) 

*  By  the  formula,  [PI.  Trig.  (413)], 

x  —  sin  x  -\-  i  sin3  x  -j-  &c. 
where  the  second  member  is  to  be  reduced  to  seconds  by  dividing  it  by  sin  1". 


CONSTANT    OF    LUNAR    PARALLAX.  683 

which  is  independent  of  ft,  and  thus  free  from  any  error  in  that 
quantity.  Small  errors  in  k  will  not  appreciably  affect  the  other 
terms  of  the  equation. 

Thus  every  pair  of  corresponding  observations  gives  an  equa- 
tion of  the  form 

0  =  n  -j-  a  sin  p  (710) 

from  which  the  parallax  p  at  the  mean  time  of  each  pair  of 
observations  could  be  derived.  But,  in  order  to  combine  all 
these  equations,  we  must  introduce  in  the  place  of  the  variable 
p  the  constant  mean  parallax,  which  is  effected  as  follows.  Let 

TC  =  the  horizontal  parallax  taken  from  the  lunar  tables  for 

the  time  £, 

KO=  the  constant  mean  parallax  of  the  tables, 
pQ  =  the  true  value  of  this  constant. 

The  form  of  the  moon's  orbit  is  well  known  :  so  that  for  any 
given  time  the  ratio  of  the  radius  vector  to  the  semi-major  axis, 
as  employed  in  the  tables,  is  to  be  regarded  as  correct;  that  is, 
the  ratio 

sin  TT 

fJL  —  —.  -  (Til) 


derived  from  the  tables,  is  to  be  regarded  as  the  ratio  between 
the  true  parallax  at  the  given  time  and  the  true  constant  :  so  that 
we  have  also 

sin  p 

p.  =  —  —  or  sm  p  =  ft,  sm  p 

smp0 

and  .the  equation  (710)  becomes 

asin;>0+£  =  0  (712) 

The  quantities  a,  ??,  and  //  being  computed  for  each  pair  of  corre- 
sponding observations,  we  thus  obtain  a  number  of  equations, 
all  involving  the  same  unknown  constant  sin  p^  which  are  then 
to  be  solved  by  the  method  of  least  squares. 

433.  The  quantities  p  and  p,  which  enter  into  the  coefficient  m, 
willbe  computed  for  an  assumed  value  of  the  compression  of  the 
earth.  But,  in  order  to  see  the  effect  of  the  compression,  we 


684  CONSTANT    OF    LUNAR    PARALLAX. 

may  isolate  the  terms  which  involve  it,  as  follows.     ^Teglecting 
the  fourth  powers  of  the  eccentricity  e,  we  have,  by  (84)  and  (83), 

p  =  1  —  %  e1  sin2  <p 
e1  sin  2  v 


2  sin  I" 

But  when  we  neglect  the  fourth  powers  of  e,  or  the  square  of  the 
compression  c,  we  have,  by  (81), 

by  which  we  obtain  the  somewhat  simpler  forms, 

p  =  1  —  c  sin2  <p 
c  sin  2  <p 

r  = 


sin  1" 
These  values  substituted  in  m  give,  by  neglecting  the  square  of  c, 

9 


m 


=  (1  —  c  sin2  p)  sin  I  C'  --  .*"  7.f  )  =F  A: 
\  sm  1"   / 

—  (1  —  c  sin2  <f>)  (sin  C'  —  c  sin  2  ^  cos  C')  =F  A; 
=  sin  C'  —  c  (sin2  ^  sin  C'  +  sin  2  <p  cos  C')  T  /: 

and,  similarly, 

ml  =  sin  C/  —  c  (sin2  $r  t  sin  C/  +  sin  2  ^j  cos  C/)  dt  k 

The  effect  of  the  compression  will  be  insensible  in  the  terms 
involving  sin3  p,  in  which  we  may  take 

m3  =  (sin  :'  q=  A')3  m,3  =  (sin  C/  ±r  A')3 

and  the  same  approximation  is  allowable  in  the  term  in  -J-.      If 

then  we  make  these  substitutions  in  (708),  we  obtain  the  follow- 
ing expanded  equation  : 

0  -  [(*!-  T)™  -  (6,-  6)-]  sin  1"+  [(sin  ^  *)3+  (sin  d'^*)']  ^ 
*-  sin  1"  [(sin  C'rp  *)  (7»-  <)  +  (sin  £'=!=*)  (7\  -  <)] 


(sin  f  '  +  sin  ^') 
—  c  //  sin  p0  [sin2  ^  sin  £  '  -{-  sin  2  0  cos  f  '  -f  sin2  ^  sin  ^'  -f  sin  2  0X  cos  ^' 


CONSTANT    OF    LUNAR    PARALLAX.  685 

If  this  equation  be  divided  by  />«,  it  may  be  expressed  under  the 
form 

0  =  n  +  x  (a  —  cb}  (713) 

where  the  notation  is  as  follows  : 


+  ^f  [  (sin  C'  HF  /0°  +  (si"  C,'  ±  *f\ 

+  ^  D>in  C'  =F  *)  (  Z1  -  t)  +  (Bin  C,  ±  *)  (  T,  -  0]  J  cos  j, 

fl  =  sin  C'  -f-  sin  C/ 

&  =  sin2  ^  sin  C'  +  sin  2  <p  cos  C'  -f-  sin2  yl  sin  C/  +  sin  2  ^  cos  C/ 

x  =  sin  p0 

It  is  here  to  be  observed  that  we  have  taken  fl  as  a  positive 
quantity  even  for  the  southern  station  :  so  that  sin  2  <pl  must  be 
taken  positively  in  computing  b. 

Let  us  now  suppose  we  have  obtained  from  a  large  number  of 
such  corresponding  observations  the  equations 

0  =  n  -J-  x  (a  —  cb} 
0  =  n'  -f-  x  (a!  —  cbf) 
0  =  n"  -f-  x  (a"  —  cb"} 
&c. 

Multiplying  these  respectively  by  a,  ar,  ar/,  &c.,  and  then  forming 
their  sum,  we  have  * 

0  =  [an]  -f  \aa\  x  —  [ab~\  ex 

where  [an]  =  an  -f  «'w'  +  &c.,  [aa]  =  aa  -j-  a'ar  +  &c.,  &c.  The 
last  term  is  very  small  :  so  that  an  approximate  value  of  x  may 
be  found  by  neglecting  it,  whence 


which  value  may  then  be  employed  with  sufficient  accuracy  in 
the  term  [a6]  ex  ;  we  thus  find  the  complete  value 

[an]       [an]  |>6] 

+ 


686  CONSTANT    OF    LUNAR    PARALLAX. 

This  is  essentially  the  method  by  which  OLUFSEN*  has  dis- 
cussed the  observations  made  by  LACAILLE  in  the  years  1751, 
1752,  and  1753,  at  the  Cape  of  Good  Hope,  and  the  correspond- 
ing observations  made  at  Paris,  Bologna,  Berlin,  and  Greenwich. 
He  found  from  all  the  observations  the  final  equation 

x  =  0.01651233  *f  0.02449201  c 
Consequently,  if  we  take  the  most  probable  value  of  c  =  — 


299.1528 
there  results 

x  =  sinp0=  0.01659420 

The  parallax  given  by  the  lunar  tables  of  BURCKHARDT  and 
DAMOISEAU  is  properly  the  sine  of  the  parallax  reduced  to  seconds. 
In  order  to  compare  this  determination  with  the  constants  of 
these  tables,  we  therefore  take 


sin  1 

The  constant  of  BURCKHARDT'S  tables  is  3420".  5;  that  of 
DAMOISEAU'S,  3420".9  ;  that  of  HANSEN'S  new  tables,  3422".  06. 
This  last  value,  which  is  derived  from  theory,  agrees  remarkably 
with  that  which  is  derived  from  direct  observation  ;  for  the 
determination  by  HENDERSON  from  corresponding  observations 
at  Greenwich  and  the  Cape  of  Good  Hope  isf  3421".  8,  and  the 
mean  between  this  and  OLUFSEN'S  value  is  3422".  3. 

434*  The  correction  of  the  moon's  parallax  may  also  be  found 
from  the  observations  of  a  solar  eclipse  at  two  places  whose  dif- 
ference of  longitude  is  great,  as  is  shown  in  the  chapter  on 
eclipses,  p.  541. 

It  is  also  possible  to  determine  the  moon's  parallax  by  com- 
paring the  different  zenith  distances  of  the  moon  observed  at  one 
and  the  same  place  between  her  rising  and  setting,  since  the 
effect  of  so  great  a  parallax  is  easily  traced  from  its  maximum 
when  the  moon  is  in  the  horizon  to  its  minimum  when  at  the 
least  zenith  distance.  But  this  very  obvious  method,  by  which, 
in  fact,  HIPPARCHUS  discovered  the  moon's  parallax,  depends  too 
much  upon  the  measurement  of  the  absolute  zenith  distances  to 
admit  of  any  great  degree  of  accuracy. 


Astronomische  Nachrichten,  No.  326.  f  Ibid,  No.  338. 


PLANETS'  MEAN  SEMLDIAMETERS.  687 


THE    MEAN    SEMIDIAMETERS    OF    THE    PLANETS. 

435.  The  apparent  equatorial  semidiameter  of  a  planet  when 
its  distance  from  the  earth  is  equal  to  the  earth's  mean  distance 
from  the  sun  is  the  constant  from  which  its  apparent  semidiameter 
at  any  other  distance  can  be  found  by  the  formula 

«  =  £          '  (715) 

in  which  SQ  is  the  mean  semidiameter  and  A  the  actual  distance 
of  the  planet  from  the  earth,  the  semi-major  axis  of  the  earth's 
orbit  being  unity.  To  find  the  value  of  SQ  from  the  values  of  s 
observed  at  different  times,  we  have  then  only  to  take  the  mean 
of  all  its  values  found  by  the  formula 

s0  -  sA  (716) 

taking  A  from  the  tables  of  the  planet  for  each  observation. 

But  here  it  is  to  be  remarked  that,  in  micrometric  measures 
of  the  apparent  diameter  of  a  planet,  different  values  will  be 
obtained  by  different  observers  or  with  different  instruments. 
The  spurious  enlargement  of  the  apparent  disc  arising  from 
imperfect  definition  of  the  limb,  or  from  the  irradiation  resulting 
from  the  vivid  impression  of  light  upon  the  eye,  will  vary  with 
the  telescope,  and  may  also  vary  for  the  same  telescope  when 
eye  pieces  of  different  powers  are  employed.  The  irradiation 
may  be  assumed  to  consist  of  two  parts,  one  of  which  is  constant 
and  the  other  proportional  to  the  semidiameter.  Those  errors 
of  the  observer  which  are  not  accidental  may  also  be  supposed  to 
consist  of  two  parts,  one  constant  and  the  other  proportional  to 
the  semidiameter;  the  first  arising  from  a  faulty  judgment  of  a 
contact  of  a  micrometer  thread  with  the  limb  of  the  planet,  the 
second,  from  the  variations  in  this  judgment  depending  on  the 
magnitude  of  the  disc  observed,  and  possibly  also  upon  any 
peculiarity  of  his  eye  by  which  the  irradiation  \sfor  him  not  the 
same  quantity  as  for  other  observers.  With  the  errors  proportional 
to  the  semidiameter  will  be  combined  also  any  error  in  the  sup- 
posed value  of  a  revolution  of  the  micrometer.  The  errors  of 
the  two  kinds  will,  however,  be  all  represented  in  the  formula 

s0  =  (s  -f  x  -|-  sy}  A  (717) 

where  x  is  the  sum  of  all  the  constant  corrections  which  the 


688  CONSTANT    OF   ABERRATION. 

observed  value  s  requires,  and  sy  is  the  sum  of  all  those  which 
are  proportional  to  s.     Now,  let  , 

s^  =  an  assumed  value  of  s0, 
ds,  =  the  unknown  correction  of  this  value 


then  the  above  equation  may  be  written 

0  =  s  A  —  $i  _j_  a:  J  +  sy  A  —  dsl 

But  syJ  will  be  sensibly  the  same  as  s^y.  It  will,  therefore,  be 
constant,  and  will  combine  with  rfsr  We  shall,  therefore,  put  z 
for  syA  —  dsv  and  then,  putting 

n  =  sJ  —  Sj 
our  equations  of  condition  will  be  of  the  form 

xA  +  z  +  n  ==  0  (718) 

from  all  of  which  x  and  2  may  be  found  by  the  method  of  least 
squares.  But  it  will  be  impossible  to  separate  the  quantity  ds^ 
from  z  ;  we  can  only  put 

(*o)  =  *i  -  2 
whereas  we  have,  for  the  true  value, 


or 

«,  =  («.)  (!  +  30  (719) 

and  then,  if  any  independent  means  of  finding  y  are  discovered, 
the  true  value  of  SQ  can  be  computed. 

THE    ABERRATION    CONSTANT   AND   THE   ANNUAL    PARALLAX    OF    FIXED 

STARS. 

436.  The  constant  of  aberration  is  found  by  (669)  when  we 
know  the  velocity  of  light  and  the  mean  velocity  of  the  earth  in 
its  orbit.  The  progressive  motion  of  light  was  discovered  by 
ROEMER,  in  the  year  1675,  from  the  discrepancies  between  the 
predicted  and  observed  times  of  the  eclipses  of  Jupiter's  satellites. 
He  found  that  when  the  planet  was  nearest  to  the  earth  the 
eclipses  occurred  about  Sm  earlier  than  the  predicted  times,  and 
when  farthest  from  the  earth  about  8m  later  than  the  predicted 


CONSTANT    OF   ABERRATION.  689 

times.  The  planet  was  nearer  the  earth  in  the  first  position  than 
in  the  second  by  the  diameter  of  the  earth's  orhit;  and  hence 
ROEMER  was  led  to  the  true  explanation  of  the  discrepancy, — 
namely,  that  light  was  progressive  and  traversed  a  distance  equal 
to  the  diameter  of  the  earth's  orbit  in  about  16m.  More  recently, 
DELAMBRE,  from  a  discussion  of  several  thousand  of  the  observed 
eclipses,  found  8™  13*.2  for  the  time  in  which  light  describes  the 
mean  distance  of  the  earth  from  the  sun.  From  this  quantity, 

which  is  denoted  by  — »  Art.  395,  we  obtain  the  aberration 
constant  by  the  formula 

k  =  - — (720) 

7    nTsinl'Vl  —  61 

Hence,  with   the   values  -^  =  493'.2,   T=  366.256,  n  =  86164, 

e  =  0.01677,  we  find  k  =  20".260.  DELAMBRE  gives  20".255, 
which  would  result  from  the  above  formula  if  we  omitted  the 


factor  i/l  —  e2,  as  was  done  by  DELAMBRE. 

On  account  of  the  uncertainty  of  the  observations  of  these 
eclipses  (resulting  from  the  gradual  instead  of  the  instantaneous 
extinction  of  the  light  reflected  by  the  satellite),  more  confidence 
is  placed  in  the  value  derived  from  direct  observation  of  the 
apparent  places  of  the  fixed  stars. 

437.  To  find  the  aberration  constant  by  observations  of  fixed  stars. — 
Observations  of  the  right  ascension  of  a  star  near  the  pole  are 
especially  suitable  for  this  purpose,  because  the  effect  of  the 
aberration  upon  the  right  ascension  is  rendered  the  more  evident 
by  the  large  factor  seed  with  which  in  (678)  the  constant  is 
multiplied.  The  apparent  right  ascension  should  be  directly 
observed  at  different  times  during  at  least  one  year,  in  which 
time  the  aberration  obtains  all  its  values,  from  its  greatest  positive 
to  its  greatest  negative  value.  If  we  suppose  but  two  observa- 
tions made  at  the  two  instants  when  the  aberration  reaches  its 
maximum  and  its  minimum,  the  earth  at  these  times  being  in 
opposite  points  of  its  orbit,  and  if  a'  and  a"  are  the  apparent 
right  ascensions  at  these  times  (freed  from  the  effects  of  the 
nutation  and  the  precession  in  the  interval  between  the  observa- 
tions), we  shall  have 

k=\(o!  —  a")  cos  d 

VOL.  I.— 44 


690  CONSTANT    OF    ABERRATION. 

But,  not  to  limit  the  observations  to  these  two  instants,  let  us 
take,  for  any  time, 

a  =  the  assumed  mean  right  ascension  of  the  star  -f-  the 

nutation  -f  proper  motion, 
a!  =  the  observed  right  ascension, 

and,  further,  let 

Aa  =  the  correction  of  the  assumed  mean  right  ascension, 
A  A:  =  the  correction  of  the  assumed  aberration  constant, 

then,  hy  (678),  we  have 

a'  =  a  -f  Aa  —  (k  -f  A&)  (cos  0  cos  e  cos  a  -f  sin  0  sin  a)  sec  (? 

or,  putting 

m  sin  M  =  sin  a 

m  cos  M  =  cos  a  cos  e 

a'  =  a  +  Aa  —  (k  -f  A  A)  m  COS  (0  —  M)  SCC  5  (721) 

Hence,  collecting  the  known  quantities,  and  putting 
a  =  —  m  cos  (0  —  M )  sec  <5 

W  =  a  -j-  ak  —  a! 

we  have  the  equation  of  condition 

flAA  -f  Aa  -f  n  =  0  (722) 

Every  observation  throughout  the  year  being  employed  to  form 
such  an  equation,  we  can  deduce  from  all  the  equations,  by  the 
method  of  least  squares,  the  most  probable  values  of  A&  and  Aa. 
Those  observations  will  have  the  greatest  weight  ki  determining 
A&,  which  are  near  the  positive  and  negative  maxima  of  the 
aberration,  where  the  coefficient  a  has  its  greatest  numerical 
values.  These  maxima  occur  for  cos  (0  —  M)  =  —  1  and  cos 
(©—  M)  =--  -f  1 ;  that  is,  for  0  =  180°  +  M  and  0  =  M . 

In  this  method  it  is  assumed  that  the  precession  and  nutation 
are  so  well  known  that  the  relative  values  of  a  are  correct,  or, 
in  other  words,  that  they  are  in  error  only  by  some  quantity 
common  to  them  all  and  denoted  by  —  Aa.  Since  the  aberra- 
tion completes  its  period  in  one  year,  the  probable  errors  of  the 
present  values  of  the  precession  and  the  nutation  constants  will 
not  become  sensible  in  the  investigation  of  the  aberration  if  the 


CONSTANT    OF   ABERRATION.  691 

observations  of  each  year  are  separately  discussed.  The  period 
of  the  leading  terms  of  the  nutation  being  only  nineteen  years, 
if  we  extend  the  observations  for  aberration  over  a  considerable 
portion  of  this  period,  it  will  be  proper  to  introduce  into  our 
equations  of  condition  a  terra  involving  the  correction  of  the 
nutation  constant,  as  will  be  seen  hereafter. 

438.  The  declinations  may  also  be  employed  for  determining 
the  aberration.     If  we  put 

8  =  the  assumed  mean  declination  -j-  the  nutation, 
A<5  =  the  correction  of  this  value, 
dr  =  the  observed  value, 

we  have,  by  (678), 

<5'  =  d  -{-  A<5  —  (k  -j-  A#)  [(sin  e  cos  d  —  cos  e  sin  d  sin  a)  cos  0 

-[-  sin  d  cos  a  sin  O] 
or,  putting 

m'  sin  M'  =  sin  d  cos  a 
m'  cos  M'  =  cos  d  sin  e  —  sin  d  cos  e  sin  a 
and  then 

a'=  —  m'  cos  (O  —  Jf ') 
ri=       d  +  a'k  —  d' 

the  equation  of  condition  is 

a'*k  -f  A<5  -f  w'  =  0  (723) 

439.  If  the  pole  star  is  employed,  which  has  a  sensible  annual 
parallax,  or  any  star  whose  parallax  is  even  suspected,  it  will  be 
proper  to  introduce  into  the  equations  of  condition  a  term  which 
represents  its  effect.     We  have,  by  (691),  introducing  the  above 
auxiliaries, 

par.  in  R.  A.  =  -f-  pr  m  sin  (O  —  M)  sec  d 
par.  in  dec.    =  -\-  pr  m'  sin  (O  —  M') 

and  hence  the  equation  of  condition  from  the  right  ascension 
will  be 

a  AA-  -f  bp  -f  Aa  -f  n  =  0  (724) 

and,  from  the  declination, 

a'Ak  Jrb'p-}-*3  +  n'=Q  (725) 


692  CONSTANT    OF    ABERRATION. 

in  which 

b  =  rm  sin  (0  —  .¥")sec  d 
b'=  rw'sin(0  — 


The  solution  of  the  equations  will  now  determine,  not  only  :*k 
and  either  AGC  or  AO,  but  also  the  parallax  p. 

440.  It  was  by  comparing  the  declinations  deduced  from  the 
meridian  zenith  distances  of  stars,  and  more  especially  of  the 
star  f  Draconis,  that  BRADLEY  discovered  the  aberration.  The 
constant  deduced  from  his  observations  by  BUSCH  is  20".  2116. 

STRUVE'S  value  of  the  constant  was  derived  from  the  declina- 
tions of  seven  stars  observed  with  a  transit  instrument  in  the 
prime  vertical.*.  The  term  representing  the  parallax  was  re- 
tained in  the  equations  of  condition,  but  merely  to  show  the 
effect  of  parallax  should  it  exist.  This  effect  was  in  every  case 
small,  and,  moreover,  for  the  different  stars  had  not  always  the 
same  sign  :  so  that  he  found  the  mean  value  of  the  constant 
from  all  the  stars  would  not  be  changed  as  much  as  0".006  by 
any  probable  parallax.  On  account  of  the  extraordinary  pre- 
cision of  this  determination  of  the  aberration,  I  here  quote  the 
individual  results  and  their  probable  errors  from  the  Astrono* 
mische  Nachrichten,  Vol.  XXI.  p.  58. 

Aberration  Probable 


Constant. 

Error. 

v  Ursce  Maj. 
i  Draconis 

20".4571 
20  .4792 

0".0303 
0  .0224 

d  Cassiopeia? 
o  Draconis 

20  .4559 
20  .4039 

0  .0462 
0  .0229 

b  Draconis 

20  .5036 

0  .0322 

P.  XIX.  371 

20  .3947 

0  .0333 

P  Cassiopeia          20  .4227         0  .0352 

whence,  having  regard  to  the  probable  errors,  the  mean  was 
found  20".4451  with  the  probable  error  0".0111  =  ^  of  a  second 
of  arc. 

Other  modern  determinations  of  the  constant  of  aberration 
agree  in  giving  a  greater  value  than  was  found  by  DELAMBRE 
from  the  eclipses  of  Jupiter's  satellites.  Thus,  LINDENAU  found 

*  See  Vol.  II.     Determination  of  the  declinations  of  stars  by  their  transits  over 
the  Prime  Vertical,  Arts.  188  et  seq. 


PARALLAX    OF    A    FIXED    STAR.  693 

\ 

from  the  right  ascensions  of  the  pole  star  k  =  20".4486,  and  the 
annual  parallax  of  the  star  =  0".1444  ;  PETERS,  from  six  hundred 
and  three  equations  of  condition,  formed  upon  the  right  ascen- 
sions of  the  pole  star,  observed  at  Dorpat  in  the  years  1822  to 
1838,  found  k  =  20//.4255,  with  the  annual  parallax  =  0".1724 ; 
LUNDAHL,  from  one  hundred  and  two  observed  declinations  of 
this  star,  found  k  =  20".5508,  and  the  parallax  ===  0".1473;  and 
PETERS,  from  two  hundred  and  seventy-nine  declinations  of 
the  same  star,  observed  with  the  Kepsold  vertical  circle  of  the 
Pulkova  Observatory,  found  k  =  20". 503,  and  the  parallax 
=  0".067*. 

The  parallax  is  so  small  a  quantity  that  the  discrepancies 
between  these  several  values  appear  to  be  relatively  great : 
nevertheless,  we  must  consider  them  as  surprisingly  small  when 
we  remember  that  all  these  determinations  rest  upon  observa- 
tions of  the  absolute  place  of  the  star.  Differential  measures  of 
the  changes  of  a  star's  place  with  the  micrometer  are  susceptible 
of  greater  refinement.  Such  a  method  I  proceed  to  give  in 
the  next  article. 

441.  To  find  the  relative  parallax  of  two  stars  by  micrometric 
measures  of  their  apparent  angular  distance. — It  was  first  suggested 
by  the  elder  HERSCHEL  that  if  the  absolute  linear  distances  of 
two  neighboring  stars  from  our  solar  system  were  very  unequal, 
their  apparent  angular  distance  from  each  other  as  seen  froni 
the  earth  would  necessarily  vary  as  the  earth  changed  its  posi- 
tion in  its  orbit.  If  one  of  the  stars  were  so  remote  as  to  have 
no  sensible  parallax,  changes  in  this  apparent  distance  (provided 
they  followed  the  known  law  of  parallax)  might  be  ascribed 
solely  to  the  parallax  of  the  nearer  star ;  and  in  any  case  such 
changes  might  be  ascribed  to  the  relative  parallax ;  that  is,  to 
the  difference  of  the  parallaxes  of  the  two  stars. 

For  the  trial  of  this  method  BESSEL  judiciously  selected  the 
star  61  Cygni,  near  which  are  two  much  smaller  stars  (at  dis- 
tances from  it  of  about  8'  and  12'  respectively),  and  from  a 
series  of  micrometric  measures  of  its  angular  distance  from 
each,  extending  through  a  period  of  more  than  a  year,  namely, 
from  August  18,  1837,  to  October  2,  1838,  obtained  the  first 
dearly  demonstrated  parallax  of  a  fixed  star.f  A  subsequent 

*Astron.  Nach.,  Vol.  XXII.  p.  119.  f  Ibid.  No.  366. 


694  PARALLAX  OF  A  FIXED  STAR. 

series  extending  from  October  10,  1838,  to  March  23,  1840, 
fully  confirmed  the  parallax,  only  slightly  increasing  its 
amount.*  The  first  series  gave  the  annual  parallax  0".3136; 
the  final  result  from  both  series  is  0".37,  with  a  probable  error 
of  ±  0".01. 

In  the  selection  of  this  star,  it  was  presumed,  in  accordance 
with  the  conception  of  HERSCHEL,  that  Gl  Cyyni,  being  between 
the  fifth  and  sixth  magnitudes,  was  much  nearer  than  the  com- 
parison stars,  which  were  both  between  the  ninth  and  tenth 
magnitudes.  A  still  stronger  presumption  in  favor  of  its 
proximity  was  found  in  its  great  proper  motion,  which  is  among 
the  greatest  yet  observed.  Moreover,  it  is  a  double  star,  and  the 
distance  of  the  middle  point  of  the  line  joining  its  two  compo- 
nents, from  each  of  the  comparison  stars,  could  be  more  accu- 
rately observed  with  the  heliometer  than  the  distance  of  two 
simple  stars. f 

The  following  is  BESSEL'S  method  of  reducing  these  observa- 
tions. 

Let  A  be  the  star  (Fig.  61)  whose  parallax  is  sought, 
(if  a  double  star,  A  will  denote  the  middle  point  be- 
tween its  components) ;  B  the  comparison  star ;  P  the 
pole  of  the  equator.  The  observations  will  be  reduced 
to  some  assumed  epoch,  as  the  beginning  of  one  of  the 
years  over  which  the  series  extends.  For  this  epoch  let 

s  =  the  distance  AB, 
P  =  the  position  angle  of  the  star  'B  at  A  —  PAB, 
a,,  3  ==  the  mean  right  ascension  and  declination  of  At 
p  =  the  relative  annual  parallax  of  A  and  B. 

If  A'  is  the  position  of  the  star  at  the  time  of  an  observation, 
as  affected  by  parallax,  it  is  easily  seen  that  the  increase  of  the 


*  Astron.  Nach.,  No.  401. 

f  The  observations  were  made  with  the  great  heliometer  of  the  Konigsberg  Obser- 
vatory. The  distance  of  two  simple  stars  is  measured  with  this  instrument  by 
bringing  the  image  of  one  star,  formed  by  one  half  of  the  object  glass,  into  coinci- 
dence with  the  image  of  the  other  star,  formed  by  the  other  half  of  the  object-glass. 
When  one  of  the  stars  is  double,  the  imnge  of  the  simple  star  is  brought  to  the 
middle  point  of  the  line  joining  the  components  of  the  double  star.  This  point  of 
bisection  can  be  more  accurately  judged  of  by  the  eye  than  the  coincidence  of  two 
superposed  images,  when  the  distance  bisected  is  within  certain  limits.  In  the 
present  case  it  was  16". 


PARALLAX  OF  A  FIXED  STAR.  695 

distance  AB  or  A'B  —  AB,  which  will  be  denoted  by  AS,  is 
given  by  the  differential  formula 

AS  =  —  Aa  cos  <5 .  sin  P  —  A<5  cos  P 

where  A<X  and  &3  are  respectively  the  parallax  in  right  ascension 
and  declination,  which  are  given  by  (691).  Substituting  these 
values,  and  then  assuming  the  auxiliaries  m  and  Jf,  such  that 

m  cos  M  =         sin  a  sin  P  -f  cos  a  sin  d  cos  P 

m  sin  M.  =  ( —  cos  a  sin  P  -f-  sin  a  sin  d  cos  P)  cos  e 

—  cos  <J  cos  Psin  e 
we  have 

AS  =  prm  cos  (0  —  M )  (726) 

The  effect  of  the  proper  motion  of  A  upon  the  distance  is 
found  as  follows.     Let 

2  =  the  angle  which  the  great  circle  in  which  the  star 

moves  makes  with  the  declination  circle, 
p  =  the  annual  proper  motion  on  the  great  circle, 
A'a,  A'<5  =  the  given  proper  motion  in  right  ascension  and 
declination,  reduced  to  the  assumed  epoch  (Art. 
379); 

then,  as  in  Art.  380,  we  find  p  and  £  by  the  formulas 

pB\nx  =  A'a  cos  d  \ 

p  COS/  =  A'(J  / 

Let  T  be  the  time  of  any  observation  reckoned  from  the  assumed 
epoch  and  expressed  in  fractional  parts  of  a  year.  In  the  above 
diagram,  if  AA'  now  represents  the  proper  motion  on  a  great 
circle  in  the  time  r,  then  AA'=rp;  and,  if  the  effect  of  the 
proper  motion  upon  the  distance  is  denoted  by  A'S,  we  have  also 
A'B  =  s  +  A'S,  A' AB  =  P—  %,  and  the  triangle  A A'B  gives 

cos  (s  -f-  A'S)  =  cos  (iy>)  cos  s  -f  sin  (rp)  sin  s  cos  (P  —  %) 

Developing  this  equation,  and  retaining  only  second  powers  of  r/>, 
we  find 

.         (r/>)2sml(P  —  x) 

±S  =  —  Tf>  COS  (P  —  fi  +  14        —Ar 

—  o 


696  PARALLAX    OF   A    FIXED    STAR. 

in  which  r  is  the  only  variable.     Taking  then  for  the  constants 

/=  — 

(728) 


the  computation  of  the  correction  for  each  observation  is  readily 
made  by  the  formula 


The  assumed  proper  motion  may,  however,  be  in  error  ;  and 
there  may  also  be  errors  in  the  observed  distances  which  are 
proportional  to  the  time  (such  as  any  progressive  change  in  the 
value  of  the  micrometer  screw,  &c.).  The  correction  for  all  such 
errors  may  be  expressed  by  a  single  unknown  correction  y  of 
the  coefficient/,  so  that  we  shall  take 

A's=(/+y)r+/'TT  (729) 

The  corrections  of  micrometric  measures  for  the  effects  of 
aberration  and  refraction*  are  treated  of  in  Vol.  II.  Chapter  X. 
We  shall,  therefore,  suppose  these  corrections  to  have  been 
applied,  and  shall  take 

s'  =  the  observed  distance  at  the  time  r,  corrected  for  differ- 
ential aberration  and  refraction, 

and  then  we  shall  have 

sr  =  s  +  AS  +  A'S  (730) 

This  equation  involves  three  unknown  quantities,  namely,  the 
distance  s,  the  parallax  involved  in  AS,  and  the  correction  y  in- 
volved in  A'S.  Let  SQ  be  an  assumed  value  of  s  nearly  equal  to 
the  mean  of  the  values  of  5',  and  put 

*  =  so  +  x 

The  substitution  of  this  in  our  equations  of  condition  will  in- 
troduce the  small  unknown  quantity  x  in  the  place  of  the  larger 

*  These  effects  are  only  differential,  and  so  small  that  the  errors  in  the  total  refrac- 
tion and  aberration  may  safely  be  assumed  to  have  no  sensible  influence.  It  is  also 
An  advantage  of  this  method  of  finding  the  parallax  of  a  star,  that  it  is  free  from 
the  errors  of  the  nutation  and  precession,  which,  being  only  changes  in  the  position 
.of  the  circles  of  reference,  have  no  effect  whatever  upon  the  apparent  distance  of 
two  stars. 


PARALLAX    OF   A    FIXED    STAR.  697 

one  s,  and  will  thus  facilitate  the  computations.  "When  all  the 
substitutions  are  made  in  the  expression  of  s',  we  obtain  the 
following  equation  : 

0  =  50  —  s'  +  /T  -f  /'rr  -|-  x  -f  ry  -f-  prm  cos  (O  —M) 
To  put  this  in  the  usual  form,  let  us  take 


c  =  rm  cos  (O  —  M) 
then  each  observation  gives  the  equation 

x  +  ry  +  cp  +  n  =  0  (781) 

and  from  all  these  equations  we  find,  by  the  method  of  least 
squares,  the  most  probable  values  of  x,  ?/,  and  p. 

In  the  determination  of  so  small  a  quantity  as  p,  it  is  neces- 
sary to  give  to  the  micrometric  measures  the  greatest  possible 
precision.  It  is  particularly  important  to  find  the  effects  of  tem- 
perature upon  the  micrometer  screw;  for  these  effects,  depending 
on  the  season,  have  a  period  of  one  year,  like  the  parallax  itself, 
and  may  in  some  cases  so  combine  with  it  as  completely  to 
defeat  the  object  of  the  observations.  At  the  time  BESSEL  pub- 
lished his  discussion  of  his  observations  on  61  Cygni,  he  had  not 
completed  his  investigations  of  the  effect  of  temperature  upon 
the  screw,  and  therefore  introduced  an  indeterminate  quantity  k 
into  his  equations  of  condition,  by  which  the  effect  upon  the 
parallax  might  be  subsequently  taken  into  account  when  the 
correction  for  temperature  was  definitively  ascertained.  This 
was  done  as  follows.  He  had  assumed  the  correction  of  a 
measured  distance  for  the  temperature  of  the  micrometer  screw 
to  be 

A"S  =  —  0".0003912  s  (t  —  49°.2) 

in  which  t  is  the  temperature  by  Fahrenheit's  scale,  and  s  is  ex- 
pressed in  revolutions  of  the  screw.  If  the  coefficient  0".  0003912 
should  be  changed  by  subsequent  investigations  to  0".  0008912 
X  (1  +  &),  each  observed  distance  would  receive  the  correction 
&"s.k,  the  quantity  n  in  the  equations  of  condition  would 
become  n  —  &"s.k,  and  the  equations  would  take  the  form 

x  +  ry  -{-  cp  —  A"S  .  k  +  n  =  0  (732) 


698  THE    NUTATION    CONSTANT. 

The  quantity  k  being  left  indeterminate,  #,  j/,  and  p  were  found 
as  functions  of  it.     The  value  of  p  was  thus  found  to  be 

—  0".3483  —  0".0533  k,  with  the  mean  error  ±  0".0141 
The  final  result  of  his  investigation  of  the  micrometer  gives* 

k  —  —  0.4893  with  the  mean  error  ±:  0.0903 
and  hence  the  corrected  value  of  the  parallax 

=  0".3744  with  the  mean  error  ±  0".0149 

If  this  result  had  been  deduced  by  comparison  with  but  one 
star,  it  could  only  be  received  as  the  relative  parallax.  BESSEL, 
however,  employed  two  stars  whose  directions  from  Q\Cygni 
were  nearly  at  right  angles  to  each  other,  and  found  nearly  the 
same  parallax  from  both ;  whence  it  follows  either  that  both 
these  stars  have  the  same  sensible  parallax,  or,  which  is  more 
probable,  that  both  are  so  distant  as  to  exhibit  no  sensible 
parallax.  This  conclusion  would  be  confirmed  if  a  comparison 
with  other  surrounding  stars  gave  the  same  parallax,  especially 
if  these  were  of  different  magnitudes ;  for  it  would  be  in  the 
highest  degree  improbable  that  all  these  stars  were  at  the  same 
distance  from  our  solar  system. 

THE   NUTATION   CONSTANT. 

442.  To  find  the  constant  of  nutation  from  the  observed  right  ascen* 
sions  or  declinations  of  a  fixed  star. — In  Art.  437  it  was  assumed 
that  the  observations  by  which  the  aberration  constant  was  de- 
termined extended  over  only  a  year  or  two  :  so  that  the  nutation 
affected  all  the  observations  by  quantities  which  differed  so  little 
that  any  error  in  the  total  nutation  would  not  sensibly  affect  the 
determination.  When  the  observations  are  extended  over  a 
longer  period,  we  may  introduce  into  the  equations  of  condition 
an  additional  term  for  the  correction  of  the  nutation.  As  before, 
let  the  mean  right  ascensions  and  declinations  be  reduced  to 
their  apparent  values  at  the  time  of  each  observation  by  means 

*  According  to  PETERS  in  the  Astron.  Nach.  Erg'dnzungs-heft,  p.  55 ;  derived  from 
BESSEL'S  Astronomische  Untersuchungen,  Vol.  I.  p.  125. 


THE   NUTATION    CONSTANT.  699 

of  an  assumed  aberration  and  nutation,  and  denote  these 
apparent  values  by  a  and  0,  and  put 

AV  =  the  correction  of  the  nutation  constant, 
a',  df  =  the  observed  right  ascension  and  declination; 
then 

a'  =  a  -|-  Aa  +  a  &k  4-  bp  +  C  Av 

d'  =  3  +  A<S  -f-  a'&k  +  ^  -j-  C'AV  (733) 

in  which,  as  before,  ACC  and  A£  are  the  corrections  of  the  star's 
mean  place,  A&  the  correction  of  the  aberration  constant,  p  the 
star's  annual  parallax,  a  and  6,  a'  and  bf  are  the  coefficients 
found  in  Arts.  437,  438,  and  439.  It  only  remains  to  express  c 
and  c'  in  terms  of  known  quantities. 

In  the  physical  theory,  it  is  shown  that  the  coefficients  of  those 
terms  of  the  nutation  formulae  (666)  which  depend  upon  2  Q? 
O  —  r,  and  Q  +  F  involve  not  only  the  nutation  constant  (the 
coefficient  of  cos  &  ),  but  also  the  precession  constant  ;  while  all 
the  other  coefficients  vary  proportionally  to  the  coefficient  of 
cos  &  .  If  we  put 

v  —  the  assumed  nutation  constant, 

v'=  the  true  "  "          =  v  +  AV 

and  if  we  express  the  relation  between  v  and  i/  by  the  equation 


and,  in  like  manner,  suppose  the  true  precession  constant  to  be 
4,  =  50".3798  (1  -f  C) 

then,  according  to  PETERS,*  the  formulae  (666),  adapted  for  any 
value  of  the  constants,  are  for  1800, 

Ae  =  (1  +  i)  [9".2231  cos  &  —  0".0897  cos  2&  -f  0".0886  cos  2<[  ] 

+  (1—  2.162  f  +  3.162  C)  [0".5510cos2Q  +  0".0093  cos  (Q  -f  r)] 
A/l  =  (1  +  i]  [—  17".2405sin  &  -f  0".2073  sin  2&  —  0".2041  sin  2<C 

-0".0677sin(C  —  O] 

+  (l—2.162i+  3.162  0[—  1  ".2694  sin2Q  +  0".1279  sin  (O  —  r) 

—  0".0213sin(O  +  ^)] 

*  Numerus  Constans  Nutationis,  p.  46.  We  have  omitted  some  terms  which  are 
inappreciable  or  of  very  short  period.  This  omission  will  not  affect  the  accuracy  of 
the  determination  of  the  quantity  v. 


700  THE    NUTATION    CONSTANT. 

The  effect  which  any  probable  correction  of  the  precession 
constant  can  have  upon  the  very  small  terms  of  these  formulae  is 
not  only  itself  very  small,  but  must  entirely  disappear  when  a 
great  number  of  observations  extending  over  a  number  of  years 
are  combined,  since  the  principal  terms  which  are  affected  by 
the  precession — namely,  those  in  2  Q — have  a  period  of  only 
six  months.  We  can,  therefore,  here  assume  £  =  0.  In  the 
formulse  for  the  nutation  in  right  ascension  and  declination  (668), 
the  terms  in  the  first  four  lines  will  be  multiplied  by  1  -j-  i,  and 
those  in  the  last  three  lines  by  1  —  2.162  £;  so  that,  if  we  denote 
by  /9  the  sum  of  the  corrections  in  R.  A.  contained  in  the  first 
four  lines,  by  f  the  sum  of  the  remaining  corrections,  and  the 
coresponding  corrections  in  dec.  by  /9'  and  7-',  we  shall  have 

Nutation  in  R.  A.  =  (1  -f  i)  £  -f  (1  —  2.162?)  r 
"          "  Dec.  =  (1  +  0  ,5'-f  (1  —  2.162 f)  -f 

or 

Nutation  in  R.  A.  =  ft  -f  r  -f  (J3  —  2.162  r  )  i 
"          "  Dec.  =  /J'-f-  /  +  (/?'  —  2.162  /)  i 

in  which  ft  -f-  f  and  /9'  -j-  f  express  the  nutation  computed 
according  to  the  assumed  constant.  Hence  we  derive 

CAV  =  cvi  =  (,3  —  2.162r)i 
C'AV  =  cr>i=  (ft'—  2.162/)  i 

and,  consequently, 

8  —  2.162  Y 


_ 
c  — 


(734) 
,  _'—  2.162/ 


which  will  be  readily  computed  for  each  observation  if  the  lunar 
nutation  (/9,  /9r)  and  the  solar  nutation  (p,  f)  have  been  separately 
computed,  as  they  usually  are.  All  the  equations  of  the  form 
(733),  whether  constructed  upon  the  right  ascensions  or  the 
declinations,  or  both,  will  then  be  treated  by  the  method  of 
least  squares,  and  the  most  probable  values  of  Aa,  A&,  p,  and  A^ 
will  be  found. 

In  this  manner  BUSCH,*  from"  BRADLEY' s  observations  of  the 
declinations  of  twenty-three  stars,  made  in  the  years  1727  to 

*  Astron.  Nach.,  No.  309. 


THE    PRECESSION    CONSTANT.  701 

1747,  and  embracing,  therefore,  a  whole  period  of  the  nutation, 
found  k  =  20//.2116,  v  ==  9".2320.  In  this  discussion  the  parallax 
of  the  stars  was  not  taken  into  account. 

•  Nearly  the  same  value  of  the  nutation  constant  follows  from 
the  more  recent  observations  at  the  Pnlkova  Observatory.  From. 
the  declinations  of  the  pole  star  observed  between  1822  and 
1838,  LUNDAHL  found  v  =  9".  2164,  and  from  the  right  ascensions 
of  the  same  star  PETERS  found  9".23G1.  The  value  9".2231, 
which  PETERS  has  adopted  in  the  Numerus  Constans  Nutationis,  is 
the  mean  of  the  three  values  found  by  BUSCH,  LUNDAHL,  and 
himself,  having  regard  to  the  weights  of  the.  several  determina- 
tions as  given  by  their  probable  errors. 

THE    PRECESSION    CONSTANT. 

443.  If  ttj,  319  and  <x2,  o2  are  the  mean  right  ascensions  and 
declinations  of  the  same  star,  deduced  from  observation  at  two 
distant  epochs  ^  and  tv  by  deducting  from  the  observed  values 
the  aberration  and  nutation,  the  annual  variations  of  the  right 
ascension  and  declination  for  the  mean  epoch  J  (^  -f-  Q  will  be 


These  annual  variations  include  both  the  precession  and  the 
proper  motion  of  the  star  ;  and,  since  both  are  proportional  to 
the  time,  it  will  be  impossible  to  distinguish  the  proper  motion 
until  the  precession  is  obtained.  If,  however,  we  suppose  that 
the  proper  motions  of  the  different  stars  observe  no  law,  or  that 
they  take  place  indiscriminately  in  all  directions,  it  will  follow 
that  the  mean  value  of  the  precession,  deduced  from  such  annual 
variations  of  a  very  large  number  of  stars,  will  be  free  from  the 
effect  of  the  proper  motions.  The  latter  are,  in  fact,  so  various 
m  direction,  although,  as  will  hereafter  be  shown,  not  entirely 
without  law,  that  this  mode  of  proceeding  must  lead  at  least  to 
an  approximation  not  very  far  from  the  truth.  Accordingly, 
from  the  a  and  6,  found  as  above  for  each  star,  we  derive  the 
m  and  n  of  Art.  374,  by  the  equations* 

*  Both  m  and  n  may  be  found  from  the  right  ascensions  alone  by  forming  equations 

of  the  form 

m  -f-  n  sin  a0  tan  (?0  =  a 

from  a  number  of  stars  and  solving  them  by  the  method  of  least  squares. 


702  THE    PRECESSION    CONSTANT. 

m  -\-  n  sin  a0  tan  d0  =  a 


n  cos  a. 


(736) 


in  which  cc0  and  dQ  are  taken  for  the  mean  epoch  J  (^  +  t^).    And 
from  the  m  and  n  thus  found  we  have,  by  (661), 


sm  s,  = 


(737) 


in  which  —  is  the  annual  luni-solar  precession  (or  the  precession 
constant),  and  —  the  annual  planetary  precession.     But  — ^-  is 

(IT  (it 

very  accurately  obtained  theoretically  by  substituting  the  known 
masses  of  the  planets  in  the  general  formula  deduced  from  the 

theory  of  gravitation :  so  that  a  value  of  the  precession  —  may 

be  derived  both  from  m  and  from  n.   In  these  formulae,  the  value 
of    ex   is   to   be    employed   as   given   by   (646)    for    the    epoch 

Having  thus  obtained  a  preliminary  value  of  the  precession, 
the  quantities  m  +  n  sin  cc0  tan  30  and  n  cos  a0,  computed  from  it 
for  each  star,  can  be  compared  with  the  a  and  b  found  by  (735), 
and  the  differences  which  exceed  the  probable  errors  of  observa- 
tion may  be  regarded  as  resulting  from  the  proper  motion  of  the 
star.  Those  stars  which  are  found  to  have  a  very  large  proper 
motion  are  then  to  be  excluded  from  the  investigation  ;  and  from 
the  remaining  ones  a  more  accurate  value  of  the  precession  will 
be  obtained. 

In  this  way,  BESSEL,  from  2300  stars  whose  places  were  deter- 
mined by  BRADLEY  for  1755  and  by  PIAZZI  for  1800,  found  the 
precession  constant  for  the  year  1750  to  be  50". 37572,  and  for 
1800,  50". 36354.*  In  this  investigation  those  stars  were  ex- 
cluded which  in  the  preliminary  computation  exhibited  .annual 
proper  motions  exceeding  0".3. 

See  also  Article  445. 


*  Fundamental   Astronomic,  p.  297,  where   the   value   50". 340499   is   found;    and 
Astron.  Nach.,  No.  92,  where  the  value  is  increased  to  50".  37572. 


MOTION   OF   THE   SUN   IN   SPACE.  703 

THE   MOTION   OF   THE   SUN   IN   SPACE. 

444.  "With  a  knowledge  of  the  precession  we  are  enabled  to 
distinguish  proper  motions  in  a  large  number  of  stars.  Upon 
comparing  these  proper  motions,  Sir  W.  HERSCHEL  was  the  first 
to  observe  that  they  were  not  without  law,  that  they  did  not 
occur  indiscriminately  in  all  directions,  but  that,  in  general,  the 
stars  were  apparently  moving  towards  the  same  point  of  the 
sphere,  or /row  the  diametrically  opposite  point.  The  latter  point 
he  located  near  the  star  A  Herculis.  This  common  apparent 
motion  he  ascribed  to  a  real  motion  of  our  solar  system,  a  con- 
clusion which  has  since  been  fully  confirmed. 

Nevertheless,  there  are  many  stars  whose  proper  motions  are 
exceptions  to  this  law :  these  must  be  regarded  as  motions  com- 
pounded of  the  real  motions  of  the  stars  themselves  and  that  of 
our  sun.  These  real  motions  must,  doubtless,  also  be  connected 
by  some  law  which  the  future  progress  of  astronomy  may 
develop  ;*  but  thus  far  they  present  themselves  in  so  many  direc- 
tions that  (like  the  whole  proper  motion  in  relation  to  the 
precession)  they  may  be  provisionally  treated  as  accidental  in 
relation  to  the  common  motion.  Hence,  for  the  purpose  of 
determining  the  common  point  from  which  the  stars  appear  to 
be  moving,  and  towards  which  our  sun  is  really  moving,  we  may 
employ  all  the  observed  proper  motions,  upon  the  presumption 
that  the  real  motions  of  the  stars,  having  the  characteristics  of 
accidental  errors  of  observation  and  combining  with  them,  will 
be  eliminated  in  the  combination.  Nevertheless,  in  order  that 
the  errors  of  observation  may  not  have  too  great  an  influence,  it 
will  be  advisable  to  employ  only  those  proper  motions  which  are 
large  in  comparison  with  their  probable  errors. 

The  direction  in  which  a  star  appears  to  move  in  consequence  of 
the  sun's  motion  lies  in  the  great  circle  drawn  through  the  star 
and  the  point  towards  which  the  sun  is  moving.  Let  this  point 
be  here  designated  as  the  point  0.  If  the  great  circle  in  which 
each  star  is  observed  to  move  were  drawn  upon  an  artificial  globe, 

*  The  law  which  we  naturally  expect  to  find  is  that  of  a  revolution  of  all  the  stars 
of  our  system  around  their  common  centre  of  gravity.  MADLER,  conceiving  that 
our  knowledge  of  the  proper  motions  is  already  sufficient  for  the  purpose,  has 
attempted  to  assign  the  position  of  this  centre.  He  has  fixed  upon  Alcyone,  the 
principal  star  of  the  Pleiades,  as  the  central  sun.  Astron.  Nach.,  No.  566.  Die 
Eigrnbcwegungen  der  Fixsternein  ihrer  Beziehung  zum  Gesammtsystem,  von  J.  H.  MADLER, 
Dorpat,  1856. 


704  MOTION    OF    THE    SUN    IN    SPACE. 

all  these  circles  would  intersect  in  the  same  point  0,  if  the  obser- 
vations were  perfect  and  the  stars  had  no  real  motion  of  their 
own.  But,  the  latter  conditions  failing,  the  intersections  which 
would  actually  occur  would  form  a  group  of  points  whose  mathe- 
matical centre  of  gravity  would,  according  to  the  theory  of  proba- 
bilities, be  the  point  from  which,  or  towards  which,  the  common 
motions  were  directed.*  Thus,  an  approximate  first  solution  might 
be  obtained  by  a  purely  graphic  process. 

Let  us  then  assume  that  an  approximate  solution  has  been 
found,  and  put 

A,  D  =  the  assumed  approximate  right  ascension  and  decli- 
nation of  the  point  0. 

It  is  then  required  to  find  a  more  exact  solution  by  determining 
the  corrections  A^.  and  AZ)  which  A  and  D  require. 
Let  P  (Fig.  62)  be  the  pole  of  the  equator,  and  S  a 
star  whose  apparent  motion  resulting  from  the  sun's 
motion  is  in  the  great  circle  OSS' .  The  angle  PSS' 
=  %,  which  this  great  circle  makes  with  the  declina- 
tion circle  (reckoned  in  the  usual  manner  from  the 
north  towards  the  east),  is  the  supplement  of  the  angle 
PSO.  Hence,  if  a  and  o  are  the  right  ascension  and 
declination  of  the  star,  and  A  the  arc  SO  joining  the 
star  and  the  point  0,  we  have,  in  the  triangle  POS, 

sin  A  sin  %  =  sin  (a  —  A}  cos  D  ) 

sin  A  cos  /  =  cos  (a  —  A)  cos  D  sin  d  —  sin  D  cos  d  j 

by  which  ^  and  ^  are  found  for  each  star. 

The  angle  ^  thus  computed  will  be  equal  to  the  observed  angle 
which  the  path  of  the  star  makes  with  the  declination  circle  only 
when  A  and  D  are  correctly  assumed.  Let  #'  be  the  observed 
angle,  or  that  which  results  from  the  equations 

P8\n/=  AaCOS<*  1 

^COS/^A^  J 

in  which  AOC  and  A/^  are  the  observed  proper  motions  in  right 
ascension  and  declination,  and  to  the  proper  motion  in  the  great 
circle.  Then,  when  %'  differs  from  ^,  the  difference  #'  —  y  is  to 
be  regarded  as  a.  function  of  the  corrections  &A  and  AD  which 
the  assumed  values  of  A  and  D  require.  The  variations  of  the 


MOTION  OF  THE  SUN  IN  SPACE.  705 

angle  £  produced  by  the  variations  of  A  and  D  will  be  found 
from  the  triangle  POS  by  the  first  differential  formulae  (47) ; 
whence 

(cos  (a  — 

+  8m("^1)C03'  ^  (740> 

Hence,  we  have  only  to  compute  for  each  star  the  values  of  £ 
and  sin  A  by  (738),  and  of  #'  by  (739),  and  then,  putting 

n  =  (y  —  •/}  sin  A 

VA  A    / 

_  cos  (a  —  A)  cos  d  sin  D  —  sin  d  cos  D 
sin  x 

sin  (a  —  A)  cos  d 
sin  A 

we  form  the  equation  of  condition, 

a .  &A  cosD  -}-  b  .  AD  -f-  n  =  0 


in  which  &A  cos  _D  and  AD  are  the  unknown  quantities.  From 
all  the  equations  thus  formed  the  most  probable  values  of  &A 
and  A!)  will  be  found  by  the  method  of  least  squares.  The 
quantity  (%  —  %')  sin  X  is  the  distance  between  the  great  circle  in 
which  the  star  really  moves  and  that  drawn  from  the  star  to  the 
point  0,  measured  at  this  point. 

In  this  manner  the  position  of  the  point  0  has  been  very  closely 
determined.  The  earlier  determinations  founded  on  a  compara- 
tively small  number  of  well  established  proper  motions  are 
those  of 

W.  HERSCHEL,  A  =  245°  53'  D  =  +  49°  38; 

and  GAUSS,        A  =  259    10  D  =  -f  30    50 

Of  the  more  recent  determinations,  the  first  in  the  order  of  time 
is  that  of  ARGELANDER.*  He  employed  390  stars,  the  proper 
motions  of  which  he  found  by  comparing  their  positions  as  deter- 
mined by  himself  for  1830f  with  those  determined  by  BESSEL  from 
BRADLEY' s  observations  for  1755. J  He  divided  these  stars  into 

*  Astron.  Nach.,  No.  363.      f  DLX  Stell.  Fix.  Positiones  Mtdix  ineunte  anno  1830. 
J  Fundamenta  Astronomise. 
VOL.  I.— 45 


706 


MOTION    OF   THE    SUN   IN   SPACE, 


three  classes  according  to  their  proper  motions,  and  found,  for 
the  epoch  1792.5, 


From 

Whose  annual  proper  motion 
was 

A  = 

D  — 

23  stars 
50     " 
319     « 

greater  than  1".0 
between  0".5  and  1  .0 
«        0  .2   "    0  .5 

256°  25M 
255     9  .7 
261   10.7 

+  38°  37'.2 
+  38    34  .3 
+  30    58  .1 

and,  combining  these   results  with   regard  to  their  respective 

weights, 

A  =  259°  51'.8  D  =  +  32°  29M 

As  supplementary  to  this  computation,  LUNDAHL  compared  147 
of  BRADLEY'S  stars  not  contained  in  ARGELANDER'S  catalogue 
with  POND'S  catalogue  of  1112  stars  for  1830,  and  found* 


A  =  252°  24'.4 


D  =  +  14°  26M 


which  ARGELANDER  combined  with  his  former  results  and  found, 
for  1800, 

A  =  257°  54'  D  =  +  28°  49' 

OTTO  STRUVE,  employing  400  stars,  mostly  identical,  however, 
with  ARGELANDER'S  and  LUNDAHL'S  stars,  and  determining  their 
proper  motions  from  the  Dorpat  observations  compared  with 
BRADLEY'S,  found,  for  1790, 


A  =  261°  21'.8 


D  =  37°  36'.0 


GALLOWAY,  from  the  southern  stars  observed  by  JOHNSON  at  St. 
Helena  and  HENDERSON  at  the  Cape  of  Good  Hope  (for  1830), 
and  by  LACAILLE  at  the  Cape  of  Good  Hope  (for  1750),  found 


A  =  260°  T 


D  =  +  34°  23' 


Finally,  MADLER,  recomputing  the  proper  motions  of  a  large 
number  of  stars,  with  the  aid  of  the  best  modern  observations, 
has  found,  for  1800,f 


From 

Whose  proper  motion  is 

A'** 

D  = 

227  stars 
663    " 
1273    « 

greater  than  0".25 
between  0".l    and  0  .25 
"       0  .04   «     0  .01 

262°  38'.8 
261    14.4 
261    32.2 

-j-  39°  25'.2 
+  37    53  .6 
+  42    21.9 

*  Astron.  Nach.,  No. 


f  Die  Eigenbewegungen  der  Fizsterne,  p.  227. 


MOTION    OF   THE    SUN    IN    SPACE.  707 

and  by  combination,  having1  regard  to  the  number  of  stars  in 
each  class, 

A  =  261°  38'  8  D  =  +  39°  53'.9 

445.  It  would  at  first  sight  seem  that  the  existence  of  any 
law  in  the  proper  motions  of  the  stars  would  vitiate  the  value 
of  the  precession  constant  found  by  BESSEL  according  to  the 
method  of  Art.  443.  Accordingly,  OTTO  STRUVE  has  attempted 
to  determine  both  the  precession  constant  and  the  motion  of  the 
solar  system  from  equations  of  condition  involving  both.  In 
order  to  accomplish  this  it  was  necessary  to  introduce  into  the 
equations  the  magnitude  as  well  as  the  direction  of  the  proper 
motions.  But  since  the  apparent  angular  motion  of  a  star,  so 
far  as  it  depends  upon  the  motion  of  our  sun,  is  a  function  of  the 
star's  distance  from  us,  it  became  necessary  also  to  make  an 
hypothesis  as  to  the  relative  distances  of  the  stars  of  different 
orders  of  magnitude.  Thus,  the  new  value  of  the  precession 
constant  given  by  him,  and  which  we  have  (provisionally)  adopted 
on  page  606,  is  also  exposed  to  the  objection  that  it  rests  upon 
an  hypothesis. 

Astronomers  have,  therefore,  been  led  to  re-examine  the 
grounds  upon  which  BESSEL'S  determination  rests.  It  is  to  be 
observed  that  the  method  which  he  employed  would  give  a  re- 
sult entirely  free  from  the  effects  of  the  sun's  motion,  if  the  stars 
employed  were  uniformly  distributed  over  the  sphere,  and  if  the 
average  distance  of  these  stars  in  all  directions  from  the  sun  were 
the  same.  MADLER,  in  the  work  above  quoted,  has  shown  that 
for  2139  stars  distributed  with  tolerable  uniformity,  BESSEL'S 
constant  gives  proper  motions  in  right  ascension  the  mean  of 
which  is  only —  0".0003.  If  now  this  quantity  were  applied  to 
BESSEL'S  value  of  m  and  the  proper  motions  again  computed, 
their  mean  would  come  out  exactly  zero.  Hence  he  concludes 
that  these  stars  fully  confirm  BESSEL'S  constant,  since  the  correc- 
tion —  0".0003  is  insignificant.  It  appears,  however,  that,  in 
drawing  this  inference  without  reservation,  he  has  left  out  of 
view  the  second  conclusion  above  stated,  that  the  average  dis- 
tance of  the  stars  on  all  sides  of  us  should  be  the  same.  For,  if 
the  sun's  motion  produces  greater  apparent  motions  in  stars  near 
to  us  than  in  those  more  remote,  a  want  of  uniformity  in  the  dis- 
tances, notwithstanding  the  equal  distribution  of  the  stars,  would 
produce  a  greater  amount  of  proper  motion  in  one  hemisphere 


708  MOTION  OF  THE  SUN  IN  SPACE. 

than  in  the  other;  and  the  aggregate  of  all  the  proper  motions, 
having  regard  to  their  signs,  would  not  be  zero. 

Since  it  is  probable  that  the  average  distance  of  stars  of  the 
same  magnitude  is  the  same  on  all  sides  of  us  (although  there  are 
not  a  few  individual  exceptions  of  small  stars  with  large  proper 
motions  and  large  stars  with  small  ones),  a  more  satisfactory 
determination  of  the  precession  constant  may  result  from  future 
investigations  in  which  not  only  all  the  stars  employed  shall  be 
uniformly  distributed,  but  those  of  each  order  of  apparent  magni- 
tude shall  be  so  distributed.  It  will  be  impossible  to  secure  this 
condition  if  the  larger  stars  are  retained ;  for  their  distribution  is 
too  unequal.  By  confining  the  investigation  to  the  small  stars, 
there  will  also  be  obtained  the  additional  advantage  that  the 
amount  of  the  proper  motions  themselves  will  probably  be  very 
small,  and  thus  have  very  little  influence  upon  the  precession 
constant,  even  if  they  are  not  wholly  eliminated.  The  formation 
of  accurate  catalogues  of  the  small  stars  is  therefore  essential  to 
the  future  progress  of  astronomy  in  this  direction. 


END    OF   VOL.  I. 


